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Progress of Theoretical Physics

Spin-3/2 Ising Model and Ashkin-Teller Model

Tomoya Horiguchi, Yasushi Honda

Published May 1, 1995• 21 citations
Publisher

Abstract

We derive a relation between a general spin-3/2 Ising model and an Ising model with two sets of Ising spins. A spin-3/2 Ising model with up-down symmetry is expressed in terms of an Ashkin-Teller model and solved exactly in some cases without assuming extended Horiguchi's condition. It turns out that the system on the square lattice shows rich critical phenomena.

DOI
10.1143/ptp.93.981

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Physical review. B, Condensed matter•May 1, 1997•11 citations
We derive a relation between a general spin-3/2 Ising model and an Ising model with two sets of Ising spins. A spin-3/2 Ising model with up-down symmetry is expressed in terms of an Ashkin-Teller model and solved exactly in some cases without assuming extended Horiguchi's condition. It turns out that the system on the square lattice shows rich critical phenomena.
The generalized Ashkin-Teller model involving both biquadratic and bilinear interactions between the Ising subsystems (α and s ) and equivalent to the anisotropic Ising model with spin 3/2 is considered. For a certain magnitude of the opposite-sign bilinear interactions along the horizontal and vertical axes of a square lattice, the exact analytic solution is obtained that describes the phase transition between the disordered (〈α 〉=〈 s 〉=〈α s 〉=0) and the correlated ordered (〈 s 〉 ≠0 and 〈α〉=〈 s 〉=0) states.
The diagrammatical approach to the exact solution of two-dimensional Ising model, developed for the system with spin 1/2 and 1, is extended to be applicable to the Ising model of spin S greater than 1/2. The exact solution for the system on the square lattice and other lattices with non-crossing interactions is obtained only when the spin is half an odd integer, and the interaction between two spins s j and s k is expressed as - J j k φ j ( s j )φ k ( s k ) in terms of one paramater J j k and odd functions φ j ( s ) and φ k ( s ) of s satisfying |φ j ( s )|=|φ k ( s )|=1 for s =- S , - S +1,· ··, and S . The exact solution on the honeycomb lattice is obtained for all S , for the cases when the interactions divided by the temperature are expressed in terms of S +1/2 or S parameters.
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.
In order to study the effect of the anisotropy parameter on the two coupled Ising models, we propose an Ashkin-Teller model (ATM) with spin variable S=1. Using an effective-field theory, we show that near the partially ordered phase PO1{\mathrm{PO}}_{1}PO1​ that appears in spin-1/2 ATM, this model presents a new partially ordered phase PO2{\mathrm{PO}}_{2}PO2​. The anisotropy parameter introduces a first-order phase transition between different phases. However, the model exhibits a rich variety of phase diagrams and higher-order critical points.