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 $p_{\text{c}}\approx 0.22$. The relaxation time is also investigated in the adiabatic case $\omega =0$ and the dynamic exponent $z\nu$ is found to increase with increasing $p$.

• Received 8 July 2021
• Accepted 31 August 2021

DOI:https://doi.org/10.1103/PhysRevE.104.034128