Ground-state energy and frustration of the Sherrington-Kirkpatrick model and related models

TL;DR

This paper computes exact ground states for the SK spin-glass model and related models, revealing the system's frustration and phase transition behavior between ferromagnetic and spin-glass states.

Contribution

It provides the first precise ground-state energy calculation for the SK model with up to 90 spins and introduces a parameter to analyze phase transitions in related models.

Findings

Ground-state energy per spin is approximately -0.7637.

Identifies a transition point at x=0.5 between ferromagnetic and spin-glass phases.

Shows the SK model's frustration and phase behavior through finite-size analysis.

Abstract

Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin $e^{\infty}_0 = - 0.7637 \pm 0.0004$ is found from the $N$ dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter $x$ is introduced, which describes the fraction of antiferromagnetic interactions in these types of models. From the $x$ dependence of finite models it is concluded, that the the SK model $(x=0.5)$ assigns a transition between a ferromagnetic state $(x<0.5)$ to a spin-glass state $(x>0.5)$.