Influence of long-range interaction on degeneracy of eigenvalues of connection matrix of d-dimensional Ising system

We examine connection matrices of Ising systems with long-rang interaction on d-dimensional hypercube lattices of linear dimensions L. We express the eigenvectors of these matrices as the Kronecker products of the eigenvectors for the one-dimensional Ising system. The eigenvalues of the connection matrices are polynomials of the d-th degree of the eigenvalues for the one-dimensional system. We show that including of the long-range interaction does not remove the degeneracy of the eigenvalues of the connection matrix. We analyze the eigenvalue spectral density in the limit L go to \infty. In the case of the continuous spectrum, for d<3 we obtain analytical formulas that describe the influence of the long-range interaction on the spectral density and the crucial changes of the spectrum.


Introduction
The Ising model is widely used in various science areas. Commonly it describes a system of interacting particles in the nodes of hypercube lattices. In the book [1], one can find a classical review of various approaches to the analysis of the Ising systems and the obtained results. Applications of the Ising model to the studies of phase transitions in solids can be found in the book [2]. The monograph [3] describes applications of this model in spin glasses and neural networks. Following the paper [4], there was a series of publications where the authors used the Ising model for training deep neural networks. A collective monograph [5] describes the relations between the Ising model and the problems of binary optimization. Useful references can also be found in [6].
In the present paper, we obtain exact expressions for the eigenvalues and eigenvectors of the Ising connection matrices on hypercube lattices taking into account interactions with an arbitrary number of neighbors. The exact eigenvalues obtained here can be used when calculating the free energy of a spin system [7], in the analysis of the role of long-range hopping in many-body localization for lattice systems of various dimensions (see [8]- [13] and references therein), and in many other applications.
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 neighbors is an approximation, which holds the better the stronger the interaction decays as a function of distance. However, for artificial spin systems with couplers, such as the ones used for quantum annealing (see for example [14]- [20]), the obtained expressions are exact.
In Section 2, we obtain exact results for the eigenvalues and eigenvectors of the Ising connection matrices with discrete spectra. In Section 3, we present the results for a continuous spectrum of the eigenvalues in the limit L → ∞ , where L is a linear size of the system. Section 4 contains discussion and conclusions.

Eigenvalues spectrum
In this Section, we obtain expressions for the eigenvalues and eigenvectors of the connection matrices of the Ising systems on hypercube lattices with an arbitrary long-range interaction and periodic boundary conditions. We, first, examine one-, two-and three-dimensional lattices and then generalize the results to the case of a hypercube.

1D Ising model
Let us consider a chain of the length L and set the distance between its nodes to be equal to one. For certainty, we suppose that L is an odd number: be an L L × symmetric matrix that describes the interactions only between spins spaced by the distance k ( 1, 2,..., k l = ). The structure of the matrix ( ) k J is as follows. The ones occupy the k -th and ( L k − )-th its diagonals which are parallel to the main diagonal and the other matrix elements are equal to zero. The central row of the matrix ( ) k J has the form where the ones are at the distance k from the center. We obtain all other rows by consequent cyclic shifts of the ( 1) l + -th row: shifting it to the left we obtain the l -th row, the right shift gives the ( 2) l + -row, and so on. For example, when All the matrices ( ) k J commute and consequently they all have the same set of the eigenvectors { } 1 . The components of the vectors α f are well-known [21]: Each matrix ( ) k J has its own set of the eigenvalues { } 1 ( ) Let ( ) w k be the constant of interaction between spins that are at the distant k from each other, where 1, 2,..., k l = . Then for the one-dimensional lattice, the interaction matrix taking account for an arbitrary long-range interaction has the form: ( For simplicity and universality, we introduce the notations, For the following calculations, the rule (3) is very useful.

2D Ising model
In this Subsection we discuss the 2D Ising model that is a system of spins in the nods of a square lattice. By ( , ) w m k we denote the constant of interaction between spins that are shifted from each other at a distance m along one of the lattice axis and at a distance k along the other axis. The connection matrix of such a system is an 2 2 L L × matrix 0 B . It is convenient to present this matrix as an L L × block matrix 0 where L L × matrices m A have the form: In Eq. (5) we set (0, 0) 0 w = and, consequently, the self-interaction is equal to zero. Note, the central block row of the matrix 0 B is ( ) all other block rows we obtain by evident cyclic shifts. We can treat a two-dimensional spin system as a set of interacting one-dimensional chains (for example, the horizontal ones.) Then the matrix 0 A describes the interactions between the spins of the one horizontal chain and the matrices m A ( 0 m ≠ ) define interactions between the spins from different chains shifted vertically by m nods. The matrix 0 B is a block Toeplitz matrix with the matrices 0 A on the main diagonal and the matrices m A on its m -th and ( Since the matrices m A commute, we can write the eigenvectors of the matrix 0 B as the Kronecker products of the eigenvectors (1): This means that we reduce the eigenvalue problem We see that the eigenvalues αβ µ are polynomials of the second degree of the eigenvalues ( ) k α λ calculated for the one-dimensional system. As example, let us examine a special case of an isotropic interaction only with the nearest neighbors (the interaction constants are (0,1) (1, 0) 1 w w = = ) and the next nearest neighbors (the interaction constant is (1,1) w ). Then the equation (8) takes the form The equation (9) repeats the result obtained previously in [22,23] where we discussed this special case.

3D Ising model
Let us discuss the three-dimensional Ising system of interacting spins that are in the nodes of a cubic lattice. By ( , , ) w n m k we denote the constant of interaction between spins shifted relative to each other by a distance n along one axis, by a distance m along the other axis, and by a distance k along the third axis. In such a system, it is convenient to write the connection 3 3 L L × matrix 0 C in the block form: where n B are 2 2 L L × matrices ( 0,1, 2..., n l = ). To obtain the matrices n B we have to generate a set of L L × matrices Since there is no self-interaction, we set (0, 0, 0) 0 w = . With the aid of the matrices (11) we generate the matrices n B : By analogy with the two-dimensional system, we can consider the three-dimensional lattice as a set of interacting planar lattices. Then the matrix 0 B describes the interactions of spins belonging to one (let us say, a horizontal) plane; the matrix n B ( 0 n ≠ ) describes the interactions between the spins from two different planes shifted with respect to each other along the vertical axis by n nodes.
As we see, the matrix 0 C has a form of a block Toeplitz matrix with the matrices 0 B at its main diagonal and the matrices n B ( 0 n ≠ ) at its n -th and ( ) L n − -th diagonals. Then, we can write the matrix 0 C as where, as usually, We see that in the three-dimensional case the eigenvalues are the polynomials of the third degree of the eigenvalues for the one-dimensional system. As an example, let us discuss a special case of the three-dimensional isotropic Ising system that is ( , , ) ( , , ) ( , , ) w n m k w k n m w m n k = = . We suppose that only the interactions with the nearest neighbors, the next nearest and the third neighbors are nonzero. We set (0, 0,1) The last expression coincides with result obtained previously in [22,23], where we examined the same special case.

Ising system on hypercube
A generalization to the case of a d-dimensional lattice is evident. Let us introduce the constants of interaction between spins 1 2 ( , ,..., ) d w k k k , where 1 k is a relative distance between the spins along the axis 1, 2 k is a relative distance between the spins along the axis 2, and so on. We generate a set of L L × matrices Then again we reduce the eigenvalue problem

Density of eigenvalue spectrum
In the previous Sections, we obtained the expressions for the eigenvalues of the connection matrices in multidimensional Ising systems, which allow us to estimate the degeneracy of their spectra. However, in the limit L → ∞ it is more efficient to pass from the discrete to continuous spectrum and analyze the spectral density ( ) P µ of the eigenvalue spectrum, where ( ) P d µ µ is the number of the eigenvalues in the interval [ , ] d µ µ µ + . In this limit we succeed in deriving analytical expressions only for one-and two-dimensional systems when we account for interactions with the nearest and the next nearest neighbors. For certainty, we suppose that the constant of interaction with the next nearest neighbor b is positive: 0 b ≥ .

Spectrum density of 1D system
Let us examine the one-dimensional system. Let (1)  . In this case the spectral density is nonzero only inside the interval In Eq. ; we show graphs for and .

Spectral density of 2D system
We analyze the two-dimensional Ising system in the simplest case supposing an isotropic interaction and accounting for interactions with the nearest and next nearest neighbors only. Let the constant of interaction with the nearest neighbors be equal to one and by   Fig. 2b presents this picture schematically.

Spectrum density for
In the case of Ising systems of higher dimensions, due to significant mathematical difficulties we were unable to obtain analytical expressions for the spectral density. However, it is possible to make some general conclusions. For example, when we account for the nearest neighbors only, we can present the spectral density ( ) d P µ for the ddimensional system as a convolution of the spectral densities of the systems of lower dimensions:

Discussion and conclusions
1 d L − . Everywhere above we examined the odd values of L . In this case, the degeneracy of the zero-valued eigenvalue was not so large. However, when L increases the number of the eigenvalues with close to zero values increases rapidly. This fact becomes evident when we turn to the case of the continuous spectrum.
In the asymptotic limit L → ∞ , it is more effective to examine not the discrete spectrum but the eigenvalue density ( ) P µ . The analysis of Section 3 shows that in the most of the discussed cases the function ( ) P µ has singularities (the logarithmic and power divergences). We have shown that the long-range interaction leads to a significant increase of the degeneracy in some regions of the continuous eigenvalue spectrum.