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Journal of Experimental and Theoretical Physics Letters

New type of exact solutions to the generalized Ashkin-Teller model and anisotropic Ising model with spin 3/2

Виктор Михайлович Розенбаум

Published Jul 1, 2003• 5 citations
Publisher

Abstract

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.

DOI
10.1134/1.1609573

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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.
Using the recursion method, we study the phase transitions of the Ashkin–Teller model on the Bethe lattice, restricting ourselves to the case of ferromagnetic interactions. The isotropic Ashkin–Teller model and the anisotropic one are respectively investigated, and exact expressions for the free energy and the magnetization are obtained. It can be found that each of the three varieties of phase diagrams, for the anisotropic Ashkin–Teller model, consists of four phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Ferro , and two partially ordered ferromagnetic phases ⟨σ⟩ and ⟨σ s ⟩, while the phase diagram, for the isotropic Ashkin–Teller model, contains three phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Baxter Phase , and the partially ordered ferromagnetic phase ⟨σ s ⟩.
Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.
The gonihedric Ising Hamiltonians defined in three and higher dimensions by Savvidy and Wegner provide an extensive, and little explored, catalogue of spin models on (hyper)cubic lattices with many interesting features. In three dimensions the kappa=0 gonihedric Ising model on a cubic lattice has been shown to possess a degenerate low-temperature phase and a first order phase transition, as well as interesting dynamical properties. The dual Hamiltonian to this may be written as an anisotropic Ashkin-Teller model and also has a degenerate low-temperature phase as a result of similar symmetries to the original plaquette action. It is possible to write an alternative dual formulation which utilizes three flavours of spins, rather than the two of the Ashkin-Teller model. This still possesses anisotropic couplings, but all the interaction terms are now four spin couplings and it acquires an additional, local gauge symmetry. We investigate this alternative dual Hamiltonian using zero temperature and mean-field methods together with Monte-Carlo simulations and discuss its properties and the relation to the Ashkin-Teller variant.
The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.