On Equilibrium Dynamics of Spin-Glass Systems

TL;DR

This paper critically examines the Sompolinsky theory of equilibrium dynamics in spin-glass systems, demonstrating its instability and proposing an alternative, stable dynamical formulation that aligns with the Parisi solution in the limit of infinite replica symmetry breakings.

Contribution

It introduces a new dynamical formulation for spin-glass systems that overcomes the instability of the Sompolinsky approach and aligns with the Parisi solution in the R→∞ limit.

Findings

Sompolinsky theory fails to produce a stable thermodynamic solution.

The new formulation reproduces the static limit consistent with the dynamic free energy threshold.

In the R→∞ limit, both formulations converge to the Parisi anti-parabolic differential equation.

Abstract

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical $2+p$ spin glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H\"orner and Sommers [Z. f\"ur Physik {\bf 92}, 257 (1993)] dynamical solution of the spherical $p$-spin spin glass model, reproducing a stable static limit that coincides, in the case of a one step Replica Symmetry Breaking Ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off-equilibrium. We formally extend our analysis to any number of Replica Symmetry Breakings $R$. In the limit $R\to\infty$ both formulations lead to the Parisi anti-parabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The new formulation does not contain the additional order parameter $\Delta$ of the Sompolinsky theory.