On the Parisi-Toulouse hypothesis for the spin glass phase in mean-field theory

TL;DR

This paper investigates the spin-glass phase of the Sherrington-Kirkpatrick model under magnetic field, analyzing the Parisi function's behavior, its scaling hypotheses, and the nature of the phase transition.

Contribution

It provides high-order series expansions of the Parisi function and rigorously confirms the order of the phase transition near the Almeida-Thouless line.

Findings

Parisi-Toulouse scaling hypotheses are mostly violated at high orders.

The phase transition on the Almeida-Thouless line is third order.

Transition smoothness varies with the magnetic field strength.

Abstract

We consider the spin-glass phase of the Sherrington-Kirkpatrick model in the presence of a magnetic field. The series expansion of the Parisi function $q(x)$ is computed at high orders in powers of $\tau=T_c-T$ and $H$. We find that none of the Parisi-Toulouse scaling hypotheses on the $q(x)$ behavior strictly holds, although some of them are violated only at high orders. The series is resummed yielding results in the whole spin-glass phase which are compared with those from a numerical evaluation of the $q(x)$. At the high order considered, the transition turns out to be third order on the Almeida-Thouless line, a result which is confirmed rigorously computing the expansion of the solution near the line at finite $\tau$. The transition becomes smoother for infinitesimally small field while it is third order at strictly zero field.