Towrad mean-field bound for critical temperature on Nishimori line

Manaka Okuyama, Masayuki Ohzeki

Published 2024-06-18Version 1

The critical inverse temperature of the mean-field approximation gives a lower bound of the true critical inverse temperature in a broad class of ferromagnetic spin models, which is called the mean-field bound for the critical temperature. In this study, we explore 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 the Nishimori line, the critical inverse temperature of the mean-field approximation is given by $\beta_{MF}^{NL}=\sqrt{1/z}$, where $z$ denotes the coordination number. Using the Griffiths inequalities on the Nishimori line, we prove that there is zero spontaneous magnetization in the high-temperature region $\beta < \beta_{MF}^{NL}/2$. In other words, the true critical inverse temperature $\beta_c^{NL}$ of the Nishimori line is always bounded by $\beta_c^{NL} \ge \beta_{MF}^{NL}/2$. Unfortunately, we did not succeed in obtaining the corresponding mean-field bound $\beta_c^{NL} \ge \beta_{MF}^{NL}$ on the Nishimori line.