Critical behavior of weakly-disordered anisotropic systems in two dimensions

TL;DR

This paper investigates the critical behavior of two-dimensional anisotropic systems with weak disorder, showing they belong to the same universality class as the 2D Ising model and supporting superuniversality in disordered models.

Contribution

It demonstrates that weakly-disordered 2D anisotropic systems described by the GATM share universality with the 2D Ising model and calculates critical exponents for related Potts models.

Findings

GATM models belong to the same universality class as 2D Ising.

Critical exponent ν for 3- and 4-state Potts models is close to Ising.

Supports superuniversality conjecture for 2D disordered models.

Abstract

The critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent $\nu$ of the correlation length for the 3- and 4-state random-bond Potts models is also calculated in a 3-loop approximation. We show that this exponent is given by an apparently convergent series in $\epsilon=c-\frac{1}{2}$ (with $c$ the central charge of the Potts model) and that the numerical values of $\nu$ are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the 3- and 4-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries.