On the two-steps relaxation of mean-field glasses: p-spin model

TL;DR

This paper derives critical slowing down exponents for mean-field spin-glass models with discontinuous transitions by linking static properties to dynamic behavior, extending Mode Coupling Theory insights.

Contribution

It introduces a method to compute critical exponents in spin-glass models using Gibbs free energy and static-in state properties, applicable to spherical and Ising cases.

Findings

Derived critical exponents for spherical and Ising p-spin models.

Generalized Mode Coupling kernels for spherical models.

Validated predictions against existing dynamic results.

Abstract

Critical slowing down dynamics of supercooled glass-forming liquids is usually understood at the mean-field level in the framework of Mode Coupling Theory, providing a two-time relaxation scenario and power-law behaviors of the time correlation function at dynamic criticality. In this work we derive critical slowing down exponents of spin-glass models undergoing discontinuous transitions by computing their Gibbs free energy and connecting the dynamic behavior to static "in-state" properties. Both the spherical and Ising versions are considered and, in the simpler spherical case, a generalization to arbitrary schematic Mode Coupling kernels is presented. Comparison with dynamic results available in literature is performed. Analytical predictions for the Ising case are provided for any $p$.