Numerical study of the SK Model in magnetic field

TL;DR

This paper numerically investigates the SK spin glass model under magnetic field, examining finite size effects, transition evidence, and the shape of the overlap distribution, highlighting the challenges and confirming theoretical predictions.

Contribution

It provides the first numerical evidence of the RSB predicted P(q) shape at large lattice size and low temperature in a non-zero magnetic field.

Findings

Strong corrections to scaling hinder precise transition detection.

Sum rules based on stochastic stability are numerically well satisfied.

The RSB shape of P(q) is observed at large lattice size.

Abstract

We study numerically the Sherrington--Kirkpatrick model as function of the magnetic field h, with fixed temperature T=0.6 Tc. We investigate the finite size scaling behavior of several quantities, such as the spin glass susceptibility, looking for numerical evidences of the transition on the De Almeida Thouless line. We find strong corrections to scaling which make difficult to locate the transition point. This shows, in a simple case, the extreme difficulties of spin glass simulations in non-zero magnetic field. Next, we study various sum rules (consequences of stochastic stability) involving overlaps between three and four replicas, which appear to be numerically well satisfied, and in a non-trivial way. Finally, we present data on P(q) for a large lattice size (N=3200) at low temperature T=0.4 Tc, where, for the first time, the shape predicted by the RSB solution of the model for non-zero magnetic field is visible.