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Journal of Physics C Solid State Physics

Two-layered Ising model with short-range two-spin interactions showing the critical behaviour of the Ashkin-Teller model

K Jungling

Published Mar 28, 1976• 3 citations
Publisher

Abstract

A two-layered Ising model is presented which maps to the Ashkin-Teller (AT) model (1943) for certain ratios of the two-spin couplings. Thus the model is able to exhibit the same critical exponents as the AT model, which are in general different from the usual Ising exponents; even continuous dependence of the critical exponents on the energy parameters is to be expected.

DOI
10.1088/0022-3719/9/6/001

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Journal of Physics C Solid State Physics•May 7, 1974•98 citations
A two-layered Ising model is presented which maps to the Ashkin-Teller (AT) model (1943) for certain ratios of the two-spin couplings. Thus the model is able to exhibit the same critical exponents as the AT model, which are in general different from the usual Ising exponents; even continuous dependence of the critical exponents on the energy parameters is to be expected.
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.