Dynamic critical exponentzin some two-dimensional models

Abstract

We discuss the current methods for determining the dynamic critical index $z$ for the dynamic universality class $n=1$, $d=2$ where the nonconserved order parameter is the only slow mode (model A). We conclude that essentially all known methods ($\ensuremath{\epsilon}$ expansions, high-temperature expansions, Monte Carlo calculations, Monte Carlo renormalization-group calculations, and the real-space dynamic renormalization method) are, at their present level of development, inconclusive. We show, in particular, that if we analyze the available high-temperature expansion data using methods similar to those used in carrying out the $\ensuremath{\epsilon}$ expansions, the resulting series is too short to extract any nonconventional value of $z$. At this level of expansion, the series is compatible with a conventional value of $z$. We show that these difficulties appear to be associated with the existence of an asymptotic dynamic critical region much narrower than the asymptotic static critical region.