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

TL;DR

This paper investigates how long-range interactions affect the eigenvalue degeneracy and spectral density of connection matrices in d-dimensional Ising systems, revealing that degeneracy persists despite long-range effects.

Contribution

It provides analytical expressions for the spectral density and demonstrates that long-range interactions do not lift eigenvalue degeneracy in these systems.

Findings

Eigenvectors are Kronecker products of 1D Ising eigenvectors.

Eigenvalues are polynomials of degree d of 1D eigenvalues.

Degeneracy remains with long-range interactions, spectral density analyzed for d<3.

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 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.