Dynamics of a coupled-mode system: Explicit analysis at orderε2

Abstract

The renormalization-group equations for critical behavior in $4-\epsilon$ dimensions are generalized to the discussion of dynamic exponents at $T=T_c$ and then applied to a situation recently considered by Hohenberg, Halperin, and Ma (HHM); namely, one in which an $n$-component primary field $\psi$ is coupled to a secondary field $\varphi$ via a ${\lambda }_0\left(\psi \psi \right)\varphi$ coupling. In this work their analysis of relaxational fields is explicitly considered at order ${\epsilon }^2$. For $n<\mathrm{}4\mathrm{}$, we find the dynamic exponents differ in one case at $O\left({\epsilon }^2\right)$ from that given by HHM. In contrast to HHM, the dynamic scaling postulate is also preserved by the second-order fixed points in all instances and detailed analysis is given. The fixed-point values of $R$, a dimensionless ratio of relaxation rates for the $\varphi$ and $\psi$ fields, in certain cases, is found to go as $R^{*}\sim \frac{1}{\epsilon }$ which is a new result. This large value where it occurs is found essential to understanding dynamic scaling. Analysis of the case of a propagating conserved $\varphi$ field also is given. It is found that its dynamic critical properties can be characterized by its damping term alone.