Thermal Conductivity of Superconducting UPt$_3$ at Low Temperatures

We study the thermal conductivity within the E$_{1g}$ and E$_{2u}$ models for superconductivity in UPt$_3$ and compare the theoretical results for electronic heat transport with recently measured results reported by Lussier, Ellman and Taillefer. The existing data down to $T/T_c\approx 0.1$ provides convincing evidence for the presence of both line and point nodes in the gap, but the data can be accounted for either by an E$_{1g}$ or E$_{2u}$ order parameter. We discuss the features of the pairing symmetry, Fermi surface, and excitation spectrum that are reflected in the thermal conductivity at very low temperatures. Significant differences between the E$_{1g}$ and E$_{2u}$ models are predicted to develop at excitation energies below the bandwidth of the impurity-induced Andreev bound states. The zero-temperature limit of the $\bf \hat c$ axis thermal conductivity, $\lim_{T\to 0} \kappa_c/T$, is \underline{universal} for the E$_{2u}$ model, but non-universal for the E$_{1g}$ model. Thus, impurity concentration studies at very low temperatures should differentiate between the nodal structures of the E$_{2u}$ and E$_{1g}$ models.