Inverse problem for Ising connection matrix with long-range interaction

TL;DR

This paper addresses an inverse problem for Ising models on hypercube lattices, determining interaction constants from known eigenvalues, and explores conditions under which a sequence can be a valid spectrum.

Contribution

It introduces a method to recover interaction constants from eigenvalues and defines restrictions for connection matrix spectra in long-range Ising systems.

Findings

Derived analytical expressions for eigenvalues with long-range interactions

Established conditions for random sequences to be connection matrix spectra

Solved the inverse problem for Ising connection matrices with periodic boundary conditions

Abstract

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of it eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.