Dynamics of symmetry-breaking long-ranged order in two-dimensional nonscalar systems

Abstract

A simple model for the dynamics of a continuous spin system in contact with a temperature bath is developed from a generalization of the Glauber model. The dependence of magnetization on time is then found in an approximation which becomes exact at temperatures far from $T_c$. For nonscalar ($N>\mathrm{}1\mathrm{}$) systems in contact with a temperature bath at $T<\mathrm{}{T}_c$, symmetry-breaking magnetization will develop and persist for a time $\tau$, after which it is destroyed by transverse fluctuations. $\tau$ is $\propto \Omega \frac{T_c}{T}$ for $d>\mathrm{}2\mathrm{}$, while $\ln \left(\tau \right)\propto \frac{T_c}{T}$ for $d=2\mathrm{}$ ($d$ is the spatial dimensionality and $\Omega$ is the volume). As $\Omega \to \infty$, $\tau$ is finite for $d=2\mathrm{}$, as required by rigourous theorems. In practice, however, a $\tau$ for $d=2\mathrm{}$ which is comparable to the $\tau$ for $d=3\mathrm{}$ systems which are large but finite is found at experimentally obtainable temperatures $T^{-1\mathrm{}}\propto T_c^{}{}_{}{}^{-1\mathrm{}}\ln \left({10}^{23}\right)$. We estimate that proportionality factors are such that "virtually macroscopic" persistence times are obtained close to $T_c$. In addition, there is evidence that the nature of the $t\gg \tau$ decay of magnetism for $d=2\mathrm{}$ systems provides a qualitative means of distinguishing subcritical from above critical systems. No virtually persistent magnetization is found for $d=1\mathrm{}$.