Tricritical dynamics near four dimensions

Phenomenological models of dynamic tricritical and $\lambda$-line behavior, with an emphasis on ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures, are found to deviate from mean-field theory below $d=4\mathrm{}$ and are studied to first order in $\epsilon =4-d$. At the tricritical point in the mixtures, the mass diffusion coefficient $D$ varies as ${\kappa }^x$, where $x=\left(\frac{1}{3}\right)\epsilon +O\left({\epsilon }^2\right)$, in disagreement with previous mode-coupling theories. The dynamic crossover scaling functions are determined, and are found to exhibit three distinct varieties of critical behavior. Although these results are consistent with present experiments, our model suffers from ambiguities similar to those discussed by Halperin, Hohenberg, and Ma for their model $C$. The dynamic properties of magnetic systems with tricritical points, such as Fe${\mathrm{Cl}}_2$, are also treated to order $\epsilon$. Ambiguities inherent in the calculations for ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures are not present in this case.