Universality and non-universality in the Ashkin-Teller model

TL;DR

This paper investigates the critical behavior of the Ashkin-Teller model with weak anisotropy and coupling, revealing continuous variation of critical temperatures and anomalous divergence in specific heat through fermionic Renormalization Group analysis.

Contribution

It provides a rigorous analysis of the Ashkin-Teller model's critical phenomena using fermionic mappings and Renormalization Group methods, highlighting the interplay of universality and non-universality.

Findings

Critical temperature difference varies continuously with coupling strength.

Specific heat diverges logarithmically at critical points, with a renormalized amplitude.

Outside the critical interval, specific heat exhibits anomalous power-law behavior.

Abstract

The Ashkin-Teller model can be formulated as a pair of 2D Ising models, interacting via a four-spin interaction. I consider the case of weak anisotropy (slight a-symmetry between the two Ising layers) and weak coupling. I show that the system admits two critical temperatures whose difference varies continuously with the strength of the coupling, scaling with an anomalous exponent as one let the a-symmetry parameter go to zero. The specific heat diverges logarithmically at the critical points (as for Ising) but the constant in front of the logarithm is renormalized by an anomalous critical exponent. The logarithmic divergence of the specific heat dominates only in an exponentially small interval around the critical temperatures and outside it is modified into an anomalous power law behaviour. The proof is based on an exact mapping of Ashkin-Teller into a model of (1+1)D interacting fermions and on the implementation of constructive Renormalization group methods on the fermionic system. This PhD thesis includes: (1) a review of the exact solution of 2D Ising in terms of Grassmann functional integrals; (2) a review of the Grassmann representation for a class of interacting Ising models, including Ashkin-Teller and the 8 vertex model; (3) a detailed discussion of the multiscale analysis of the Ashkin-Teller model, based on fermionic Renormalization Group methods, including the study of the flow of the effective coupling constants (in particular a proof of vanishing of the beta function for this and similar Luttinger-like models, based on modified approximate Ward identities, is included). This thesis is based on joint work with V. Mastropietro.