Criticality versus q in the 2+1-dimensional $Z_q$ clock model

TL;DR

This study uses Monte Carlo simulations to analyze the critical behavior of the 2+1-dimensional $Z_q$ clock model, revealing that for $q \,\geq\, 5$, its critical exponents match those of the $XY$ model, indicating a limiting behavior.

Contribution

It demonstrates that the critical exponents of the $Z_q$ clock model converge to the $XY$ model values for $q \geq 5$, showing the onset of the limiting behavior at relatively small $q$.

Findings

Critical exponents for $q \geq 5$ match the $XY$ model.

Limiting behavior occurs already at $q=5$.

Critical properties are similar to the $XY$ model for sufficiently large $q$.

Abstract

Using Monte Carlo simulations we have studied the $d=3$ $Z_q$ clock model in two different representations, the phase-representation and the loop/dumbbell-gas (LDG) representation. We find that for $q \ge 5$ the critical exponents $\alpha$ and $\nu$ for the specific heat and the correlation length, respectively, take on values corresponding to the case $q\to \infty$, where $\lim_{q \to \infty} Z_q = 3DXY$ model, i.e. in terms of critical properties the limiting behaviour is reached already at $q=5$.