Critical dynamics of isotropic antiferromagnets using renormalization-group methods:T<TN

We analyze a simple model appropriate for the description of the critical dynamics of isotropic antiferromagnets in the ordered phase. We use renormalization-group methods and mode-coupling ideas to analyze the various correlation functions of interest below the Néel temperature. The transverse correlation functions are dominated by spin waves in agreement with the predictions of hydrodynamics in the appropriate $q\xi \ll 1$ limit. We find that the positions of the spin-wave peaks for the transverse magnetization and staggered magnetization correlation function differ significantly as $q\xi$ increases. The spin-wave peaks persist as $q\xi$ is increased in the transverse staggered magnetization, but become quickly overdamped as $q\xi$ increases for the transverse magnetization correlation function. These results are consistent with our previous calculations at $T=T_N$ where we found damped spin waves in the staggered magnetization correlation function, but not in the magnetization correlation function. Our most interesting result is that as a result of strong coupling of transverse spin waves into the longitudinal modes, hydrodynamics breaks down in treating the longitudinal magnetization correlation functions. This breakdown, first discovered by Villain using a mode-coupling approach, is manifested in the wave-number-dependent spin-diffusion coefficient going as $q^{-\frac{1}{2}}$ in three dimension for small $q$. The treatment of the longitudinal order-parameter correlation function is difficult within an $\epsilon$ expansion. We discuss these difficulties.