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

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.