Boundary criticality via gauging finite subgroups: a case study on the clock model

TL;DR

This paper explores how gauging finite subgroups of symmetries in certain models leads to unconventional critical points, revealing deep connections between symmetry breaking, topological phases, and anomalies in various dimensions.

Contribution

It introduces a framework for understanding boundary criticality via gauging finite subgroups, with explicit examples in clock models and analysis of associated anomalies.

Findings

Gauging finite subgroups can produce deconfined quantum critical points.

Explicit analysis of $ ext{Z}_2$ gauging in clock models on 1D and 2D lattices.

Identification of anomaly structures similar to those in $SU(2)$ gauge theories with $ heta= ext{pi}$.

Abstract

Gauging a finite Abelian normal subgroup $\Gamma$ of a nonanomalous 0-form symmetry $G$ of a theory in $(d+1)$D spacetime can yield an unconventional critical point if the original theory has a continuous transition where $\Gamma$ is completely spontaneously broken and if $G$ is a nontrivial extension of $G/\Gamma$ by $\Gamma$. The gauged theory has symmetry $G/\Gamma \times \hat{\Gamma}^{(d-1)}$, where $\hat{\Gamma}^{(d-1)}$ is the $(d-1)$-form dual symmetry of $\Gamma$, and a 't Hooft anomaly between them. Thus it can be viewed as a boundary of a topological phase protected by $G/\Gamma \times \hat{\Gamma}^{(d-1)}$. The ordinary critical point, upon gauging, is mapped to a deconfined quantum critical point between two ordinary symmetry-breaking phases ($d =1$) or an unconventional quantum critical point between an ordinary symmetry-breaking phase and a topologically ordered phase ($d\ge 2$) associated with $G/\Gamma$ and $\hat{\Gamma}^{(d-1)}$, respectively. Order parameters and disorder parameters, before and after gauging, can be directly related. As a concrete example, we gauge the $\mathbb{Z}_2$ subgroup of $\mathbb{Z}_4$ symmetry of a 4-state clock model on a 1D lattice and a 2D square lattice. Since the symmetry of the clock model contains $D_8$, the dihedral group of order 8, we also analyze the anomaly structure which is similar to that in the compactified $SU(2)$ gauge theory with $\theta =\pi$ in $(3+1)$D and its mixed gauge theory. The general case is also discussed.