Toward mean-field bound for critical temperature on Nishimori line

TL;DR

This paper investigates the possibility of a mean-field bound for the critical temperature in Ising spin glass models on the Nishimori line, establishing a lower bound but not the conjectured equality.

Contribution

The study proves a lower bound for the critical inverse temperature on the Nishimori line using Griffiths inequalities, advancing understanding of phase transitions in spin glasses.

Findings

Bounded the true critical inverse temperature by half the mean-field estimate.

Established zero spontaneous magnetization in the high-temperature region.

Identified limitations in extending the bound to the full mean-field estimate.

Abstract

The critical inverse temperature of the mean-field approximation establishes a lower bound of the true critical inverse temperature in a broad class of ferromagnetic spin models. This is referred to as the mean-field bound for the critical temperature. In this study, we explored the possibility of a corresponding mean-field bound for the critical temperature in Ising spin glass models with Gaussian randomness on the Nishimori line. On this line, the critical inverse temperature of the mean-field approximation is given by $\beta_{MF}^{NL}=\sqrt{1/z}$, where $z$ is the coordination number. Using the Griffiths inequalities on the Nishimori line, we proved that there is zero spontaneous magnetization in the high-temperature region $\beta < \beta_{MF}^{NL}/2$. In other words, the true critical inverse temperature $\beta_c^{NL}$ on the Nishimori line is always bounded by $\beta_c^{NL} \ge \beta_{MF}^{NL}/2$. Unfortunately, we have not succeeded in obtaining the corresponding mean-field bound $\beta_c^{NL} \ge \beta_{MF}^{NL}$ on the Nishimori line.