Toward mean-field bound for critical temperature on Nishimori line

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 ${\beta }_{MF}^{NL}=\sqrt{1\mathrm{/}z}$, 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 $\beta <{\beta }_{MF}^{NL}\mathrm{/}2$. In other words, the true critical inverse temperature ${\beta }_c^{NL}$ on the Nishimori line is always bounded by ${\beta }_c^{NL}\ge {\beta }_{MF}^{NL}\mathrm{/}2$. Unfortunately, we have not succeeded in obtaining the corresponding mean-field bound ${\beta }_c^{NL}\ge {\beta }_{MF}^{NL}$ on the Nishimori line.