Renormalization-group treatment of the critical dynamics of superfluid helium, the isotropic antiferromagnet, and the easy-plane ferromagnet

Abstract

Phenomenological models with "planar-spin" and "antiferromagnetic" dynamics are introduced, and their critical behavior is analyzed using renomalization-group methods. Dynamic scaling is shown to hold for these models to all orders in $\epsilon =4-d$, and the dynamic exponents are expressed entirely in terms of static exponents, in agreement with earlier phenomenological and mode-coupling theories. The magnitudes of the diverging transport and kinetic coefficients are expressed purely in terms of static properties and of universal constants which are calculated to second order in $\epsilon$. Matching conditions are proved between the characteristic frequencies above and below $T_c$, and the corresponding universal amplitude ratios are calculated to second order in $\epsilon$. The principal applications are to liquid helium and the Heisenberg antiferromagnet RbMn${\mathrm{F}}_3$, where the experimental exponents and amplitudes both agree reasonably well with theory. In the case of liquid helium ("asymmetric" case with $\alpha <0\mathrm{}$) the asymptotic critical behavior is somewhat masked by correction terms, due to the slow approach of the spectific heat to its finite value at $T_{\lambda }$. These correction terms are analyzed in detail, and a proposal is made for extracting the true asymptotic behavior from the data. The effects of other conserved fields such as the mass density and momentum in helium, and the energy density in the magnet, are considered, and shown to be irrelevant for the critical behavior of the order parameter.