Competing anisotropies and phase transitions in the $q$-state clock model with a $p$-fold crystalline field

Abstract

We study the two-dimensional $q$-state clock model in the presence of an additional $p$-fold symmetry-breaking crystalline field using Monte Carlo simulations. While the pure clock model exhibits Berezinskii--Kosterlitz--Thouless (BKT) transitions for sufficiently large $q$, the effect of competing discrete anisotropies on this topological phase remains nontrivial. We show that even weak crystalline fields qualitatively modify the phase diagram by suppressing the BKT phase and inducing transitions to states with true long-range order. The resulting behavior depends sensitively on the interplay between the intrinsic $\mathbb{Z}_q$ symmetry and the imposed $\mathbb{Z}_p$ anisotropy. In particular, in the six-state clock model for $p=2$ we observe qualitatively different scenarios depending on the sign of the field: a single transition for $h_2>0$ and a two-step ordering process for $h_2<0$ with an intermediate ordered phase. For $p=3$, the system exhibits a direct transition consistent with three-state Potts criticality. These results demonstrate that the phase structure cannot be inferred from symmetry considerations alone, but is governed by the competition between distinct locking mechanisms. Our findings provide a discrete counterpart to the multi-frequency sine-Gordon description of generalized $XY$ models and illustrate how additional anisotropies reshape topological phase transitions in two dimensions.