Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T=0

TL;DR

This paper analyzes the stability of the Parisi solution for the Sherrington-Kirkpatrick model near zero temperature by examining the Hessian spectrum, revealing stability features and a connection to dynamic approaches.

Contribution

It provides a detailed spectral analysis of the Hessian near T=0, showing the collapse of replica symmetry breaking bands and establishing a link with dynamic time-scale methods.

Findings

Hessian spectrum maintains full RSB structure near T=0 for small x

Only two eigenvalues (null and positive) survive as T approaches zero

Static susceptibility matches the dynamic susceptibility in the static limit

Abstract

To test the stability of the Parisi solution near T=0, we study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit $T\ll 1$ two regions can be identified. In the first region, for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region $T\ll x \leq 1$ as $T\to 0$, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for $T\ll 1$. In the limit $T\to 0$ the width of the first region shrinks to zero and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.