Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick

TL;DR

This paper introduces a novel perturbation expansion method for mean-field spin-glass models that bypasses the replica trick, enabling direct calculation of physical quantities across all temperatures.

Contribution

It develops a replica-trick-free perturbation expansion for the free-energy functional with a continuous order-parameter, applicable to all mean-field spin-glass models.

Findings

First correction beyond one-level replica-symmetry breaking explicitly evaluated

Applicable at all temperatures, including zero temperature

Method simplifies calculations of physical quantities in spin glasses

Abstract

The full mean-field solution of spin glass models with a continuous order-parameter function is not directly available and approximate schemes must be used to assess its properties. The averaged physical quantities are to be represented via the replica trick and the limit to zero number of replicas is to be performed for each of them. To avoid this we introduce a perturbation expansion for a mean-field free-energy functional with a continuous order-parameter function without the need to refer to the replica trick. The expansion can be used to calculate all physical quantities in all mean-field spin-glass models and at all temperatures, including zero temperature. The small expansion parameter is a difference between the continuous order-parameter function and the corresponding order parameter from the solution with one level of replica-symmetry breaking. The first correction beyond the approximation with one level of replica-symmetry breaking is explicitly evaluated in the glassy phase of the Sherrington-Kirkpatrick model.