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Journal of Physics A Mathematical and General

Exactly soluble Ising models with anisotropic interactions and arbitrary external magnetic field

Pedro L. Garrido, Joaquín Marro

Published May 11, 1987• 4 citations
Publisher

Abstract

The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.

DOI
10.1088/0305-4470/20/7/028

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The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.
The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.
We have used the two-dimensional Ising model with a limited number of rows, but with the coordination number of four for each site, to set up the transfer matrix for the model. From the solution of such a matrix, the exact thermodynamic properties have been obtained for the model with a definite number of rows, n . We have solved the matrix for n ≤ 7 and n ≤ 10 in the presence and absence of a magnetic field, respectively. On the basis of such solutions, we have proposed an analytical expression for the partition function of the model with any number of rows in the absence of a magnetic field. The proposed expression becomes more accurate when n is larger, in such a way that it becomes very accurate for n ≥ 8 and is exact for n → ∞. Our results show that the singularity of the specific heat occurs only for the model with infinite number of rows. In the presence of a magnetic field, the solution to the matrixes is too complicated to propose a general analytical expression for the partition function of the model with any number of rows. However, our exact solution for the model with n = 4, 5, 6, and 7 reveals an important point that such results are independent of n when the field−spin interaction energy is almost equal to or larger than that of the spin−spin interaction.
A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved ``Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.
Two Ising systems are investigated with the two extended-mean-field boundary conditions recently considered by Bolton and Johnson. The systems are NNN spins on a line with boundary spins on either side and at each end. The exact solutions are obtained by the use of the modified-transfer-matrix method. A general formula for the critical temperature is found for each system. The critical temperature is found to take an asymptotic form similar to that obtained by Ferdinand and Fisher for the two-dimensional planar infinite Ising lattice. The thermodynamic limits of the results are discussed.