Phase transition dynamics in the three-dimensional field-free $\pm J$ Ising model

By using frustration-preserving hard-spin mean-field theory, we investigated the phase transition dynamics in the three-dimensional field-free $\pm J$ Ising spin glass model. As the temperature $T$ is decreased from paramagnetic phase at high temperatures, with a rate $\omega=-dT/dt$ in time $t$, the critical temperature depends on the cooling rate through a clear power-law $\omega^a$. With increasing antiferromagnetic bond fraction $p$, the exponent $a$ increases for the transition into the ferromagnetic case for $p<p_\text{c}$, and decreases for the transition into the spin glass phase for $p>p_\text{c}$, signaling the ferromagnetic-spin glass phase transition at $p_\text{c}\approx0.22$. The relaxation time is also investigated, at adiabatic case $\omega=0$, and it is found that the dynamic exponent $z\nu$ increases with increasing $p$.