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.