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

TL;DR

This study investigates phase ordering dynamics in 3D XY and Ising models, revealing slow growth, the Mpemba effect, and their experimental implications through Monte Carlo simulations.

Contribution

It demonstrates the occurrence of the Mpemba effect in spin models with different initial conditions and explores slow domain growth in the 3D XY model.

Findings

Characteristic length grows as t^{0.15} at T_f=0 in 3D XY model.

Higher initial temperatures T_s lead to faster approach to equilibrium, indicating the Mpemba effect.

In 2D Ising, the Mpemba effect occurs only near zero initial magnetization; in 3D, it appears regardless of initial magnetization distribution.

Abstract

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.