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Journal of Physics A Mathematical and General

Saddle-point mean-field theory for long-range Ising spin models in terms of the eigenvalues and eigenvectors of the interaction matrix

Andrew Canning

Published Sep 21, 1992• 7 citations
Publisher

Abstract

The mean-field saddle-point theory for Ising spin models with long-range interactions is rewritten in terms of the eigenvalues and eigenvectors of the interaction matrix. This gives a natural division of long-range models into two classes: those where the rank of the interaction matrix is finite (in the limit N to infinity ); and the saddle-point integral for the partition function can be directly evaluated (subject to several other weaker conditions) and those where the interaction matrix has a divergent rank. In the latter case the saddle-point integral method cannot be directly applied to the partition function. In the former case of the simply solvable models there are natural order parameters associated with the eigenvectors of the non-zero eigenvalues which characterize the system. It will be shown that models in this class are also described by Curie-Weiss mean-field type equations.

DOI
10.1088/0305-4470/25/18/008

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The mean-field saddle-point theory for Ising spin models with long-range interactions is rewritten in terms of the eigenvalues and eigenvectors of the interaction matrix. This gives a natural division of long-range models into two classes: those where the rank of the interaction matrix is finite (in the limit N to infinity ); and the saddle-point integral for the partition function can be directly evaluated (subject to several other weaker conditions) and those where the interaction matrix has a divergent rank. In the latter case the saddle-point integral method cannot be directly applied to the partition function. In the former case of the simply solvable models there are natural order parameters associated with the eigenvectors of the non-zero eigenvalues which characterize the system. It will be shown that models in this class are also described by Curie-Weiss mean-field type equations.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
The Ising model with ferromagnetic interactions that decay as $\frac{1}{{r}^{\ensuremath{\alpha}}}$ is analyzed in the nonextensive regime $0<~\ensuremath{\alpha}<~d$, where the thermodynamic limit is not defined. In order to study the asymptotic properties of the model in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit (NNN being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the d=1d=1d=1 case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.
The partition function of the Ising model with NNN spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site lll by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
A mean-field replica-type theory is presented for long-range Ising spin-glasses whose interactions can be written in the form J ij = J ij I J ij SK , where J SK is a quenched Gaussian interaction, as in the SK spin-glass, and J I is a deterministic matrix which defines the structure or inhomogeneity of the model and must have a finite rank in the thermodynamic limit. The spin-glass model proposed by Franz, Parisi and Virasoro ( Europhys. Lett. , 17 (1992) 5) can be written in this form, with J I having rank 2, and is used as a case study to illustrate our theory. In the context of replica theory we also consider the stable states of this model, which appear continuously as the temperature is lowered.
The Ising model at magnetic field H=0 is one of the most important exactly solved models in statistical mechanics, solved on the square lattice in 1944 by Onsager, and later by others on the triangular lattice; its magnetic susceptibility at H=0 continues to be an unsolved aspect of the model, however. The susceptibility can either be viewed as a sum over all correlation functions of the integrable model at H=0, or else as a second derivative of the free energy of the non-integrable Ising model in a field. Therefore, its analytic properties, though derived from the non-integrable model, can be studied through series expansion of the integrable correlation functions. We begin this process by analyzing the first four terms in the form factor expansion of the diagonal correlation functions, and after summing over the diagonal form factor expansion, the first four terms of the diagonal susceptibility expansion. We have been able to reduce the form factor and susceptibility expansion terms, given as multi-dimensional integrals, to closed-form functions in all cases. Under the limit of H going to infinity with the interaction energy E going to minus infinity, the isotropic Ising model becomes a hard particle lattice gas model, with exclusion of nearest neighbor lattice sites. In this limit, the triangular lattice Ising model becomes the exactly solved hard hexagon model, while the square lattice becomes the non-integrable hard squares model. We study in detail these models for finite lattice sizes in order to understand the differences between integrable and non-integrable models. We consider the partition functions zeros and their density for different boundary conditions, and find notable differences in the density which is attributed to an extra factorization in the transfer matrices of hard hexagons which is absent in hard squares. We also study the special point at fugacity z=-1 of hard squares where all eigenvalues of the transfer matrix are equimodular and where the grand partition function's value depends on boundary conditions. | 300 pages