Mean-field theory is exact for Ising spin glass models with Kac potential in non-additive limit on Nishimori line

TL;DR

This paper proves that the mean-field theory accurately describes Ising spin glass models with Kac potential on the Nishimori line, confirming Mori's conjecture in this specific setting across all temperatures and dimensions.

Contribution

The paper provides a rigorous proof of Mori's conjecture for the exactness of mean-field theory in certain spin glass models on the Nishimori line, extending understanding in non-additive regimes.

Findings

Mean-field theory is exact on the Nishimori line at any temperature and dimension.

The proof utilizes the Gibbs-Bogoliubov inequality.

Open problems remain for symmetric interactions in higher dimensions at low temperatures.

Abstract

Recently, Mori [Phys. Rev. E 84, 031128 (2011)] has conjectured that the free energy of Ising spin glass models with the Kac potential in the non-additive limit, such as the power-law potential in the non-additive regime, is exactly equal to that of the Sherrington-Kirkpatrick model in the thermodynamic limit. In this study, we prove that his conjecture is true on the Nishimori line at any temperature in any dimension. One of the key ingredients of the proof is the use of the Gibbs-Bogoliubov inequality on the Nishimori line. We also consider the case in which the probability distribution of the interaction is symmetric, where his conjecture is true at any temperature in one dimension but is an open problem in the low-temperature regime in two or more dimensions.