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

TL;DR

<3-5 sentence high-level summary> The paper develops a lattice-based framework for gauging finite Abelian higher-form symmetries and their subgroups in quantum spin models in $d=2$ and $d=3$, showing that gauging acts as an isomorphism between bond algebras and yields dual higher-group symmetries with potential mixed 't Hooft anomalies. It demonstrates that mixed anomalies manifest as symmetry fractionalization among the participating higher-form symmetries and uses this to constrain phase diagrams and map gapped phases across dual descriptions, including nonsymmetric and symmetry-enriched topological orders. In $d=2$ the authors connect short-range entangled and symmetry-enriched gapped phases to dual Higgs/deconfined cases, while in $d=3$ they reveal dualities between topological orders via 1-form symmetry gauging, including self-dual points that can host non-invertible higher-form symmetries. Collectively, the work provides a concrete lattice realization of global categorical symmetry concepts, enabling nonperturbative diagnostics of anomalies and dualities and offering a roadmap for realizing and manipulating exotic symmetry structures in quantum spin systems.

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.

Figures (19)

• Figure 1: The action of symmetry operators via conjugation can be understood as topological linking in spacetime. 0-form symmetry operators act on all of space therefore, conjugating a local operator with the symmetry operators amounts to linking a point with a co-dimension-1 sphere in the spacetime picture. Similarly 1-form symmetries are generated by co-dimension-1 operators in space. These act on lines by conjugation. Intersection on a time slice becomes linking in the spacetime picture.
• Figure 2: The gauging-related duality realized as an isomorphism of bond algebras, furnishes a mapping of order parameters. The figure depicts a mapping of operators between a bond algebra $\mathfrak B$ with 0-form symmetry $\mathsf G_{(0)}$ and a gauged bond algebra $\mathfrak B^{\vee}$ with a $(d-1)$-form symmetry $\mathsf G^{\vee}$. A 0-form symmetry operator restricted to an open ball-like region $S^{(d)}$ maps to an operator charged under the dual $(d-1)$-form symmetry at the boundary of $S^{(d-1)}$. A 0-form charge bilinear located at points $x_{\mathsf i}$ and $x_{\mathsf f}$ maps to $(d-1)$-form symmetry operator, i.e., a non-unique line connecting the $x_{\mathsf i}$ and $x_{\mathsf f}$.
• Figure 3: Symmetry-twisted boundary conditions with $\mathsf g\in\mathsf G$ are implemented by inserting a symmetry defect $\mathcal{U}_{\mathsf g}[\Sigma]$ on a non-contractible cycle along time. Above illustrates a spacetime $M = S^1\times T^2=T^3$ (opposite sides of the cube are identified) and the symmetry defect wraps the $x-t$ cycle. Any bond-operator crossing the defect, will be transformed by $\mathsf g$. An equivalent way to achieve this is to couple the symmetry $\mathsf G$ to a background connection $A$, with a holonomy $\oint_\gamma A = \mathsf g(\gamma)$ along the cycle $\gamma$ dual to $\Sigma$. Any operator parallel transported along $\gamma$ will experience a symmetry-twist.
• Figure 4: Parallel transport along a curve $L$ from $\mathsf v_1$ to $\mathsf v_k$ on $M_{d, \triangle}$ of the $\mathbb Z_n$ connection $a$.
• Figure 5: The figure depicts the operators that are used to define the Gauss operators in (a) d=1, (b) d=2 and (c) d=3 dimensions.