$E_{1g}$ model of superconducting UPt$_3$

The phase diagram of superconducting UPt$_3$ is explained in a Ginzburg-Landau theory starting from the hypothesis that the order parameter is a pseudo-spin singlet which transforms according to the $E_{1g}$ representation of the $D_{6h}$ point group. We show how to compute the positions of the phase boundaries both when the applied field is in the basal plane and when it is along the c-axis. The experimental phase diagrams as determined by longitudinal sound velocity data can be fit using a single set of parameters. In particular the crossing of the upper critical field curves for the two field directions and the apparent isotropy of the phase diagram are reproduced. The former is a result of the magnetic properties of UPt$_3$ and their contribution to the free energy in the superconducting state. The latter is a consequence of an approximate particle-hole symmetry. Finally we extend the theory to finite pressure and show that, in contrast to other models, the $E_{1g}$ model explains the observed pressure dependence of the phase boundaries.