Anomalous universality in the anisotropic Ashkin-Teller model

TL;DR

This paper rigorously analyzes the anisotropic Ashkin-Teller model, confirming the universality of critical exponents in the weakly interacting case while revealing anomalous nonuniversal behavior near the isotropic limit.

Contribution

It provides an explicit expression for the specific heat, proves the universality conjecture in the weakly interacting case, and uncovers anomalous scaling behaviors near the isotropic point.

Findings

Universality of critical exponents holds in the weakly interacting case.

Nonuniversal critical indexes appear near the isotropic limit.

The critical points and specific heat behavior are precisely characterized.

Abstract

The Ashkin-Teller (AT) model is a generalization of Ising 2-d to a four states spin model; it can be written in the form of two Ising layers (in general with different couplings) interacting via a four-spin interaction. It was conjectured long ago (by Kadanoff and Wegner, Wu and Lin, Baxter and others) that AT has in general two critical points, and that universality holds, in the sense that the critical exponents are the same as in the Ising model, except when the couplings of the two Ising layers are equal (isotropic case). We obtain an explicit expression for the specific heat from which we prove this conjecture in the weakly interacting case and we locate precisely the critical points. We find the somewhat unexpected feature that, despite universality holds for the specific heat, nevertheless nonuniversal critical indexes appear: for instance the distance between the critical points rescales with an anomalous exponent as we let the couplings of the two Ising layers coincide (isotropic limit); and so does the constant in front of the logarithm in the specific heat. Our result also explains how the crossover from universal to nonuniversal behaviour is realized.