Spherical spin-glass - Coulomb gas duality: solution beyond mean-field theory

TL;DR

This paper introduces an exact solution for a spherical spin-glass model by mapping it onto a Coulomb gas and analyzing special functions, providing insights beyond traditional mean-field approaches.

Contribution

It presents a novel method to solve the spherical spin-glass model exactly using Coulomb gas mapping and Painlevé functions, extending beyond mean-field theory.

Findings

Exact free energy obtained via Painlevé τ^{IV} function.

Thermodynamics matches classical results by Kosterlitz, Thouless, and Jones.

Method potentially applicable to systems with replica symmetry breaking.

Abstract

We present an alternate solution of a Gaussian spin-glass model with infinite ranged interactions and a global spherical constraint at zero magnetic field. The replicated spin-glass Hamiltonian is mapped onto a Coulomb gas of logarithmically interacting particles confined by a logarithmic single particle potential. The precise free energy is obtained by analyzing the Painlev\'e $\tau^{IV} [n]$ function in the $n\to 0$ limit. The large $N$ thermodynamics exactly recovers that of Kosterlitz, Thouless and Jones \cite{ktjonesPRL}. It is hoped that the approach here can be extended to apply to systems beyond the spherical model, particularly those in which destabilizing terms lead to replica symmetry breaking.