Residual discrete symmetry of the five-state clock model

TL;DR

This paper investigates the symmetry properties of the five-state clock model and finds it does not exhibit true Kosterlitz-Thouless transition behavior, unlike the six-state model, due to residual discrete symmetry.

Contribution

The study reveals that the five-state clock model lacks full U(1) symmetry in the high-temperature phase, challenging previous assumptions about its phase transition nature.

Findings

Five-state clock model does not have fully continuous U(1) symmetry at high temperatures.

The upper transition in the five-state model is not a true KT transition.

Six-state clock model exhibits genuine KT behavior, unlike the five-state model.

Abstract

It is well-known that the $q$-state clock model can exhibit a Kosterlitz-Thouless (KT) transition if $q$ is equal to or greater than a certain threshold, which has been believed to be five. However, recent numerical studies indicate that helicity modulus does not vanish in the high-temperature phase of the five-state clock model as predicted by the KT scenario. By performing Monte Carlo calculations under the fluctuating twist boundary condition, we show that it is because the five-state clock model does not have the fully continuous U(1) symmetry even in the high-temperature phase while the six-state clock model does. We suggest that the upper transition of the five-state clock model is actually a weaker cousin of the KT transition so that it is $q \ge 6$ that exhibits the genuine KT behavior.