Critical dynamics near theλtransition inHe3-He4mixtures

The critical dynamics of the $\lambda$ transition in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures are studied by means of renormalized field theory applied to the model of Siggia and Nelson. A diagonal representation for the equations of motion is introduced, which greatly simplifies the computations in two-loop order. A universal connection is found with the asymptotic critical dynamics of pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ in all orders of perturbation theory. The observable critical dynamics of helium mixtures are dominated by nonuniversal crossover effects which can be properly described only within a nonlinear renormalization-group approach. The theory is applied to explain the observable critical and precritical temperature dependence of the mass diffusion $D$, of the thermal conductivity $\kappa$, of the thermal diffusion ratio $k_T$, and of the dynamic structure factor for $T\ge T_{\lambda }\mathrm{}\left(X\right)\mathrm{}$. Recent experimental data for the transport coefficients at the molar ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ concentration $X=0.05\mathrm{}$ by Gestrich and Meyer are used to identify the nonuniversal parameters of the theory in the range $X\ll 1$. Consistency with the dynamics of pure ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ ($X=0\mathrm{}$) is verified. Predictions without adjustable parameters are made for the dynamic structure factor and the transport coefficients in very dilute mixtures. The Siggia-Kawasaki problem concerning the leading $X$ dependence of $\kappa \left(T_{\lambda }\right)$ in the $X\to 0$ limit is resolved. It is demonstrated that Siggia's prediction $\kappa \left(T_{\lambda }\right)\sim X^{-1\mathrm{}}$ is correct but not observable. Theoretical extrapolations to $X>\mathrm{}0.05\mathrm{}$ without adjustable parameters are presented and compared with measured transport coefficients at $X=0.11\mathrm{} \mathrm{and} 0.15$. The overall agreement is satisfactory. Deviations of order 15% exist with the thermal conductivity $\kappa$ at $X=0.15\mathrm{}$. This may be attributed to dynamic effects arising from the singular specific heat and mass susceptibility, which are not included in the present analysis.