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Journal of Physics A Mathematical and General

Large-Nlimit of theXY-Ashkin-Teller model: solution in two dimensions

Giuseppe Gonnella, A. Lamura

Published Jul 29, 1999• 0 citations
Publisher

Abstract

We introduce a vector model coupled to an N -colour Ashkin-Teller model and solve its phase diagram in the large- N limit in two dimensions. It is shown that the transition line starts from the axis of Ising couplings with a behaviour which can be critical or first order depending on the strength of the four-spin coupling and of the coupling between spin and vector variables. Below a negative value of the four-spin coupling the transition is always continuous.

DOI
10.1088/0305-4470/32/32/302

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We introduce a vector model coupled to an N -colour Ashkin-Teller model and solve its phase diagram in the large- N limit in two dimensions. It is shown that the transition line starts from the axis of Ising couplings with a behaviour which can be critical or first order depending on the strength of the four-spin coupling and of the coupling between spin and vector variables. Below a negative value of the four-spin coupling the transition is always continuous.
The NNN-color Ashkin-Teller model is solved exactly in the large-NNN limit in two dimensions. The phase diagram is found. It is shown that for positive four-spin coupling the transition is first order while for negative four-spin coupling the transition is continuous and Ising type. The specific heat near the second-order transition is calculated and it is found to be finite at the transition because of large corrections to scaling. The latent heat and correlation length at the first-order transition are also calculated.
We study a model Hamiltonian consisting of NNN Ising models coupled pairwise through a four-spin interaction K4{K}_{4}K4​. When N=2N=2N=2 this model is the well-known Ashkin-Teller model which shows nonuniversal critical behavior in two dimensions in the neighborhood of K4=0{K}_{4}=0K4​=0. To see if this behavior persists for $N\ensuremath{\ne}2$ we perform a first-order-perturbation expansion around the decoupling point in two dimensions. As an aid in interpreting the results of this perturbation expansion we have determined the phase diagram of the system through mean-field theory and Monte Carlo studies in both two and three dimensions for N=3N=3N=3. The results show that N=2N=2N=2 is special because the coupling between Ising models is marginal over a range of values of K4{K}_{4}K4​. We discuss the effect of the coupling K4{K}_{4}K4​ for $N\ensuremath{\ne}2$.
Magnetic properties and phase diagrams of a spin-1/2 anisotropic Ashkin-Teller model in a transverse field are studied using the mean field theory. On the one hand, it is found that there exists a critical value of the transverse field Ωc/J1 above which the partially ordered phase 〈σS〉 appears at very low temperature and sufficiently large values of the four-spin interaction coefficient J4/J1. Such result is not obtained in the Ω/J1 = 0 case. On the other hand, the multicritical behaviour is also studied as a function of the transverse field Ω/J1 for selected values of the spin interaction coefficient J2/J1. However, depending on the transverse field value, and a fixed J2/J1 value, four varieties of phase diagrams are found. General formulae are established for an arbitrary coordination number q. The phase diagrams and the behaviour of the transverse and longitudinal magnetisations are given for q = 4.