A variational approach to Ising spin glasses in finite dimensions

TL;DR

This paper develops a hierarchical variational method using the replica approach and Parisi ansatz to approximate the free energy of finite-dimensional Ising spin glasses, improving accuracy with larger clusters.

Contribution

Introduces a hierarchical variational approximation for finite-dimensional Ising spin glasses that interpolates between the SK model and the true system, providing rigorous bounds.

Findings

Provides rigorous bounds for the quenched free energy.

Bounds become more precise with larger clusters.

Method bridges SK model and finite-dimensional spin glasses.

Abstract

We introduce a hierarchical class of approximations of the random Ising spin glass in $d$ dimensions. The attention is focused on finite clusters of spins where the action of the rest of the system is properly taken into account. At the lower level (cluster of a single spin) our approximation coincides with the SK model while at the highest level it coincides with the true $d$-dimensional system. The method is variational and it uses the replica approach to spin glasses and the Parisi ansatz for the order parameter. As a result we have rigorous bounds for the quenched free energy which become more and more precise when larger and larger clusters are considered.