Sherrington-Kirkpatrick model near $T=T_c$: expanding around the Replica Symmetric Solution

TL;DR

This paper develops a fourth-order expansion of the free energy functional for the SK model near the critical temperature, revealing the existence and temperature range of the Full Replica Symmetry Broken solutions around different reference states.

Contribution

It provides explicit FRSB solutions near T_c by expanding around both the RS and paramagnetic solutions, clarifying their temperature ranges of validity.

Findings

FRSB solution around RS exists for 0.549... ≤ T ≤ 1

FRSB solution around paramagnetic state exists for 0.915... ≤ T ≤ 1

Expansion up to fourth order captures key features of the FRSB phase

Abstract

An expansion for the free energy functional of the Sherrington-Kirkpatrick (SK) model, around the Replica Symmetric SK solution $Q^{({\rm RS})}_{ab} = \delta_{ab} + q(1-\delta_{ab})$ is investigated. In particular, when the expansion is truncated to fourth order in. $Q_{ab} - Q^{({\rm RS})}_{ab}$. The Full Replica Symmetry Broken (FRSB) solution is explicitly found but it turns out to exist only in the range of temperature $0.549...\leq T\leq T_c=1$, not including T=0. On the other hand an expansion around the paramagnetic solution $Q^{({\rm PM})}_{ab} = \delta_{ab}$ up to fourth order yields a FRSB solution that exists in a limited temperature range $0.915...\leq T \leq T_c=1$.