Eigenvalues of Ising connection matrix with long-range interaction

doi:10.1016/j.physa.2020.124929

Physica A: Statistical Mechanics and its Applications

Volume 558, 15 November 2020, 124929

https://doi.org/10.1016/j.physa.2020.124929 Get rights and content

Highlights

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We examine multidimensional Ising systems on hypercube with long-range interaction.
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We obtain exact formulas for eigenvalues of connection matrices.
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We analyze both periodic and free boundary conditions.

Abstract

We examine multidimensional Ising systems on hypercube lattices and calculate analytically the eigenvalues of their connection matrices. We express the eigenvalues in terms of spin–spin interaction constants and the eigenvalues of the one-dimensional Ising connection matrix (the latter are well known). To do this we present the eigenvectors as Kronecker products of the eigenvectors of the one-dimensional Ising connection matrix. For periodic boundary conditions, it is possible to obtain exact results for interactions with an arbitrary large number of neighboring spins. We present exact expressions for the eigenvalues for two- and three-dimensional Ising connection matrices accounting for the first five coordination spheres (that is interactions up to next-next-next-next nearest neighbors). In the case of free-boundary systems, we show that in the two and three dimensions the exact expressions could be obtained only if we account for interactions with spins of not more than first three coordination spheres.

Introduction

The Ising model remains in a focus of scientific interest for decades. Partially it is caused by convenience, which this model offers for testing new methods for calculation of the free energy and critical characteristics. In the same time, in spite of a seemingly straightforward formulation of the problem, in some cases the scientists could not find its rigorous solutions. Book [1] contains a review of different approaches to the solution of the Ising problem and the obtained results. There is a number of books, where applications of the model to analysis of various physical problems are collected. Namely, in the book [2] phase transitions in solids are discussed in terms of the Ising model; the model’s applications to spin glasses and neural network theory can be found in the monograph [3]; and the collective monograph [4] describes the link between the Ising model and optimization problems.

In the present paper, we obtain exact expressions for the eigenvalues of the Ising connection matrix on the hypercube lattices with long-range interaction. We analyzed both the periodic and free boundary conditions. Everywhere we assume an isotropic interaction. Our paper generalize the analysis presented in [5].

The proximity between spins is commonly described by the expressions “the nearest neighbor”, “the next nearest neighbor”, “the next-next nearest neighbor”. However, since we consider interactions with spins that are far enough from the given spin, in place of repeating the “next” prefixes we use the concept of coordination spheres [6], [7]. Let us organize an increasing sequence of various distances from a given spin to all other spins. Then a $k$ th coordination sphere includes all the nodes whose distance to the given node takes the $k$ th place in this sequence. Clearly, all the spins belonging to a coordination sphere interact equally with the given spin. Let $w_{k}$ be the interaction constant with the spins belonging to the kth coordination sphere. When the value of $k$ is not too large, the common notations are convenient. Namely, the nearest neighbors belong to the first coordination sphere, the next nearest neighbors to the second coordination sphere, the next-next nearest neighbors to the third coordination sphere. In what follows we use the coordination sphere numbers as well as the common notations.

In Section 2, we discuss the one-dimensional Ising model in detail. We introduce matrices $J (k)$ that describe the interactions with only the spins of the $k$ th coordination sphere. In the case of periodic boundary conditions, all the matrices $J (k)$ are circulant matrices, all of them have the same set of the eigenvectors and their eigenvalues are well known. For the one-dimensional Ising model, this allows us to easily obtain, in the most general form, the eigenvalues of its connection matrix $A_{1} = \sum_{k} w_{k} J (k)$ accounting for an arbitrary number of the coordination spheres. On the contrary, if we assume free boundary conditions, the matrices $\tilde{J} (k)$ even do not commute. (In what follows we use the tilde symbol to denote characteristics related to free boundary conditions.) As a result, it is not possible to express the eigenvalues of the matrix ${\tilde{A}}_{1} = \sum_{k} w_{k} \tilde{J} (k)$ in terms of the eigenvalues of the individual matrices $\tilde{J} (k)$ . In this case, we only know the eigenvalues ${\{{\tilde{λ}}_{i}\}}_{1}^{n}$ of the matrix $\tilde{J} (1)$ that accounts for the interaction $w_{1}$ with the nearest neighbors.

In Section 3, we examine the two- and three-dimensional Ising models with periodic boundary conditions. In the two-dimensional case, we account for the interactions with the spins of the first and second coordination spheres. In the three-dimensional case, we add the interaction with the spins of the third coordination sphere. We restrict our consideration to these relatively simple models due to the following reasons. Firstly, here we present in detail our method, which allows us to define the eigenvalues of the multidimensional Ising connection matrix in terms of the interaction constants and the eigenvalues of the one-dimensional connection matrix. This method is rather cumbersome, so it is convenient to use simple examples. Secondly, as we show in Section 4, the obtained results are fully applicable also to the case of free boundaries. In Section 4, we also show that for free boundary conditions it is impossible to extend these results to the case of more neighbors.

Finally, in Section 5 we develop another method for calculation of the eigenvalues of the multidimensional Ising model for periodic boundary conditions. We base our calculation on representing the eigenvectors of the multidimensional problem as the Kronecker product of the eigenvectors of the one-dimensional Ising model. The method is especially effective when we account for a large number of coordination spheres. As an example, for the planar and cubic lattices we obtain exact expressions for five coordination spheres.

The obtained results can be useful when analyzing spectral densities of the multidimensional Ising systems and their dependencies on the parameters of the spin–spin interactions. Our exact expressions for the eigenvalues can be applied when studying optical transitions and/or in calculations of the free energy of spin systems. In addition, they can be significant in the analysis of the role of long-range hopping in many-body localization for lattice systems of various dimensions (see [8] and references therein). For natural spin systems, the interaction constants are typically determined by the distances between the spins. Then truncating the number of interactions by accounting only for a finite number of coordination spheres is an approximation, which holds the better the stronger the coordinate dependence of the interaction. However, for artificial spin systems with couplers, such as the ones used for quantum annealing (see for example [9]), our results with a finite number of coordination spheres can be viewed as exact.

Section snippets

1D-Ising model

In this Section, we examine the one-dimensional Ising model that describes a linear chain of n connected spins $s_{i} = \pm 1$ , $i = 1, \dots, n$ . In Section 2.1, we assume periodic boundary conditions. This implies that the chain of spins forms a closed loop: the $n$ th spin of the chain is the left nearest neighbor of the first spin, the second last ( $n - 1$ )th spin turns out to be the next nearest left neighbor of the first spin, and so on. In this case, some rigorous results were previously obtained [10], [11], [12].

2D- and 3D-Ising systems with periodic boundary conditions: simple case

In this Section, we analyze relatively simple models of two-dimensional and three-dimensional Ising systems with periodic boundary conditions. In two dimensions, we take into account the interactions with the spins from the first and the second coordination spheres (Section 3.1). When considering the three-dimensional model, we account for the first, second, and third coordination spheres (Section 3.2).

Let us agree on notations. For the lattices of both dimensions, the spins of the first

2D-Ising model

All the arguments of Section 3.1 also hold when we examine the case of free boundaries. All we need is to replace the matrix $J (1)$ by the matrix $\tilde{J} (1)$ , the eigenvalues $λ_{i} \equiv λ_{i} (1)$ (2.2) by the eigenvalues ${\tilde{λ}}_{i} \equiv {\tilde{λ}}_{i} (1)$ (2.5), and the eigenvectors $f_{i}$ (2.3) by the eigenvectors ${\tilde{f}}_{i}$ (2.5). Then repeating all the calculations we obtain an analog of Eq. (3.8): ${\tilde{μ}}_{i j} = w_{1} \cdot ({\tilde{λ}}_{j} + {\tilde{λ}}_{i}) + w_{2} \cdot {\tilde{λ}}_{i} {\tilde{λ}}_{j}, i, j = 1, 2, \dots, n .$ Now we can easily explain why for a planar free-boundary Ising system it is not possible to generalize

2D and 3D Ising models with periodic boundary conditions: general case

In this Section, we develop another method for calculation of the eigenvalues of the connection matrix for the multidimensional Ising model. Computationally, it is more efficient. The method represents the eigenvectors of the multidimensional problem as the Kronecker products of the eigenvectors of the one-dimensional Ising connection matrix (2.3). For periodic boundary conditions, this procedure is always correct.

A planar lattice is a sublattice of a cubic lattice. Consequently, sometimes the

Discussion and conclusions

We obtained the analytical expressions for the eigenvalues of the connection matrices of 2D- and 3D-Ising systems taking into account the interactions with the spins of several coordination spheres. We expressed the eigenvalues of the multidimensional connection matrix in terms of the interaction constants ${\{w_{r}\}}_{1}^{k}$ and the eigenvalues ${\{λ_{i}\}}_{1}^{n}$ of the one-dimensional Ising model with the interactions of the nearest neighbors. The expressions for the eigenvalues ${\{λ_{i}\}}_{1}^{n}$ are well known for periodic boundary

CRediT authorship contribution statement

L.B. Litinskii: Conceptualization, Methodology, Writing - original draft. B.V. Kryzhanovsky: Investigation, Formal analysis, Writing & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by State Program of Scientific Research Institute for System Analysis, Russian Academy of Sciences , project no. 0065-2019-0003.

The authors are grateful to Dr. Inna Kaganova and Dr. Marina Litinskaya for helpful discussions and their help when preparing this paper.

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Eigenvalues of Ising connection matrix with long-range interaction

Highlights

Abstract

Introduction

Section snippets

1D-Ising model

2D- and 3D-Ising systems with periodic boundary conditions: simple case

2D-Ising model

2D and 3D Ising models with periodic boundary conditions: general case

Discussion and conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Physica A

Exactly Solved Models in Statistical Mechanics

Introduction to Phase Transitions and Critical Phenomena

Introduction to the Theory of Spin-Glasses and Neural Networks

New Optimization Algorithms in Physics

Energy Band Theory