Phase Ordering in a few O(n) Symmetric Models: Slow Growth, Mpemba Effect and Experimental Relevance

We study phase ordering dynamics in the three-dimensional nonconserved XY model, via Monte Carlo simulations, for quenches from paramagnetic phase to certain final temperatures $T_f$ within the ferromagnetic region of the phase diagram. The growth in the system occurs via annihilation of vortex and anti-vortex pairs, cores of which, in the three dimensional system geometry, join from different planes, on which the spins lie, to form line defects. In the long-time limit, the associated characteristic length scale, $\ell(t)$, appears to grow with time $(t)$ approximately as $t^{0.15}$, for $T_f=0$. The exponent is much smaller, like in the zero temperature intermediate time ordering in the three dimensional Ising model, than $1/2$, the expected value, that is realized for quenches to $T_f$ value that is sufficiently larger than zero. We carry out quenches from different starting temperatures, $T_s$, that lie above the critical temperature $T_c$. It is observed that the systems with higher $T_s$ approach the final equilibrium faster. This resembles the puzzling Mpemba effect. We present similar results also from the simulations of the two- and three- dimensional Ising model. In the case of the 2D Ising model, we show that the Mpemba effect is observed only if the starting magnetization is restricted to a value close to zero. In $d=3$, on the other hand, for both the models, the effect appears even if the initial configurations at a given $T_s$ are chosen from the full distribution of magnetization. Thus, our results are of much experimental relevance.