Diagonalization of replicated transfer matrices for disordered Ising spin systems

TL;DR

This paper introduces a new method for solving eigenvalue problems of replicated transfer matrices in disordered spin systems, transforming large matrices into integral operators, applicable to complex models including neural networks and small world magnets.

Contribution

The authors develop an alternative formalism that simplifies the eigenvalue problem for disordered spin systems with complex interactions, extending known results to new models.

Findings

Method accurately recovers known results for the Ising chain with randomness.

Successfully applies to models with complex long-range interactions.

Numerical simulations confirm the validity of the approach.

Abstract

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g. $1+\infty$ dimensional neural networks and `small world' magnets. Numerical simulations confirm our predictions satisfactorily.