Phase transitions in $q$-state clock model

TL;DR

This paper systematically studies the phase transitions of the 2D $q$-state clock model using mean-field theories, confirming the existence of three phases for large $q$ and characterizing the nature of the phase transitions.

Contribution

The authors develop higher-order mean-field theories to better understand the phase diagram and transition types in the 2D $q$-state clock model, especially for large $q$.

Findings

For large $q$, three phases are confirmed: ferromagnetic, BKT, and paramagnetic.

The higher-temperature transition is of BKT type, while the lower-temperature transition is a large-order SSB transition.

Higher-order mean-field theory improves phase characterization by estimating neighbor spin correlations.

Abstract

The $q-$state clock model, sometimes called the discrete $XY$ model, is known to show a second-order (symmetry breaking) phase transition in two-dimension (2D) for $q\le 4$ ($q=2$ corresponds to the Ising model). On the other hand, the $q\to\infty$ limit of the model corresponds to the $XY$ model, which shows the infinite order (non-symmetry breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in 2D. Interestingly, the 2D clock model with $q\ge 5$ is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the $q$-state clock model systematically. Our MF calculations reaffirm that, for large $q$, there are three phases: (broken) $\mathbb{Z}_q$ symmetric ferromagnetic phase at the low temperature, emergent $U(1)$ symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking (SSB) type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.