Clock model interpolation and symmetry breaking in O(2) models

TL;DR

This paper introduces an extended clock model that interpolates between integer values of q and explores its phase transitions and symmetry breaking properties using Monte Carlo and tensor network methods, revealing crossover behaviors and phase diagram features.

Contribution

It defines a novel extended q-state clock model for noninteger q and investigates its phase transitions and symmetry breaking using advanced numerical methods.

Findings

Noninteger q clock model exhibits crossover and second-order phase transition.

Phase diagram of extended-O(2) model outlined, connecting XY and clock models.

Models serve as testbeds for symmetry breaking in quantum simulators.

Abstract

The $q$-state clock model is a classical spin model that corresponds to the Ising model when $q=2$ and to the $XY$ model when $q\to\infty$. The integer-$q$ clock model has been studied extensively and has been shown to have a single phase transition when $q=2$,$3$,$4$ and two phase transitions when $q>4$.We define an extended $q$-state clock model that reduces to the ordinary $q$-state clock model when $q$ is an integer and otherwise is a continuous interpolation of the clock model to noninteger $q$. We investigate this class of clock models in 2D using Monte Carlo (MC) and tensor renormalization group (TRG) methods, and we find that the model with noninteger $q$ has a crossover and a second-order phase transition. We also define an extended-$O(2)$ model (with a parameter $\gamma$) that reduces to the $XY$ model when $\gamma=0$ and to the extended $q$-state clock model when $\gamma\to\infty$, and we begin to outline the phase diagram of this model. These models with noninteger $q$ serve as a testbed to study symmetry breaking in situations corresponding to quantum simulators where experimental parameters can be tuned continuously.