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

Abstract

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 dth 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 → ∞. In the case of the continuous spectrum, for d ⩽ 2 we obtain analytical formulas that describe the influence of the long-range interaction on the spectral density and the crucial changes of the spectrum.

Eigenvalues of Ising connection matrix with long-range interaction

“…Let ( , , ) w n m k be a constant of interaction between spins shifted with respect to each other by a distance n along the first axis, by a distance m along the second axis, and by a distance k along the third axis. When the interaction is anisotropic, there are In paper [3], we showed that the ( ) has to be hold. As in the two-dimensional problem, the cases of anisotropic and isotropic interactions differ significantly.…”

“…We succeeded to obtain analytical expressions for the eigenvalues of the above-described Ising connection matrices. For the d -dimensional system, the eigenvalues are polynomials of the degree d in the eigenvalues for the one-dimensional system with longrange interaction (see [2,3]). The coefficients of these polynomials are the constants of interaction between spins.…”

“…In papers [1][2][3], we calculated eigenvalues of Ising connection matrices defined on d -dimensional hypercube lattices ( 1, 2,3...…”

“…It eigenvectors αβγ F are the Kronecker products of the eigenvectors( ) ,..., L α β γ = .The vectors αβγ F constitute a full set of the eigenvectors of any connection matrix of the three-dimensional Ising system and they do not depend on the type of the interaction constants k = . Let us write down the eigenvalues of the matrix 0 C obtained in[3]: -dimensional vector Μ whose coordinates are the eigenvalues (19): we can rewrite the set of equations (19) in the vector form: 0,1) (0, 0,1) ...(0, , ) (0, , ) (1, 0, 0) (1, 0, 0) ... (1, , ) (1, , ) .... ( , 0, 0) ( , 0, 0) .... ( , , ) ( , , ). equation (22) allows us to solve the inverse problem and calculate the interaction constants ( , , ) w n m k that define the given set of the eigenvalues { } , , matrix.…”

Inverse Problem for Ising Connection Matrix with Long-Range Interaction

“…In papers [1][2][3], we calculated eigenvalues of Ising connection matrices defined on d-dimensional hypercube lattices (d = 1, 2, 3 . .…”