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Journal of Physics Conference Series

Eigenvalues and eigenvectors of Ising model on hypercube

Boris Kryzhanovsky, Leonid B. Litinskii

Published Feb 1, 2021• 0 citations
Publisher

Abstract

Abstract For a multidimensional Ising model, we expressed eigenvalues of the connection matrix in terms of the spin-spin interaction constants and trigonometric polynomials. In such systems, the eigenvectors are the Kronecker products of the well-known eigenvectors for the one-dimensional case. When boundary conditions are periodic, it is possible to obtain rigorous expressions for the eigenvalues when there is an arbitrary long-range interaction in the system.

DOI
10.1088/1742-6596/1745/1/012038

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Abstract For a multidimensional Ising model, we expressed eigenvalues of the connection matrix in terms of the spin-spin interaction constants and trigonometric polynomials. In such systems, the eigenvectors are the Kronecker products of the well-known eigenvectors for the one-dimensional case. When boundary conditions are periodic, it is possible to obtain rigorous expressions for the eigenvalues when there is an arbitrary long-range interaction in the system.
Abstract The connection matrix of the Ising model on a ddd-dimensional hypercube is investigated. In addition to the interactions between the nearest neighbors, the interactions between the next-nearest neighbors are taken into account. For such a matrix, the exact relations for the eigenvalues and eigenvectors, which are reasonably simply expressed through the corresponding characteristics of the one-dimensional Ising model, are obtained. Both periodic and free boundary conditions are considered.
Собственные значения гамильтониана многомерной модели Изинга выражаются через константы меж-спинового взаимодействия и собственные значения одномерного гамильтониана. Для периодических граничных условий точные аналитические выражения могут быть получены практически для произвольного дальнодействия. For the Hamiltonian of multidimensional Ising system we express the eigenvalues in terms of the spin-spin interaction constants and the well-known eigenvalues of the one-dimensional spin Hamiltonian. In the case of periodic boundary conditions, we can obtain exact analytic expressions for eigenvalues of the multidimensional Hamiltonian for an arbitrary long-range interaction.
We analyze a finite one-dimensional Ising system with periodic boundary conditions taking into account an arbitrary long-range interaction. We examine a discrete spectrum of eigenvalues of the spin connection matrix and a spectrum density of a continuous distribution obtained in the limit L→∞L \to \infty L→∞ (L is the linear size of the system). We apply our results to particular cases of long-range interactions decreasing with distance.
The 16th-order transfer matrix of the three-dimensional Ising model in the particular case n = m = 2 (n × m is number of spins in a layer) is specified by the interaction parameters of three basis vectors. The matrix eigenvectors are divided into two classes, even and odd. Using the symmetry of the eigenvectors, we find their corresponding eigenvalues in general form. Eight of the sixteen eigenvalues related to odd eigenvectors are found from quadratic equations. Four eigenvalues related to even eigenvectors are found from a fourth-degree equation with symmetric coefficients. Each of the remaining four eigenvalues is equal to unity.
The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.
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.