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

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 $q=5$.

Minor corrections; journal ref added