Exact solution of the Ginzburg-Landau equations for the upper critical field of multicomponent superconductors

We present an exact solution of the problem of the upper critical field of an ${\mathit{E}}_1$ or ${\mathit{E}}_2$ superconductor. This resolves a long-standing difficulty in the theory of these materials, since only variational methods have been available up to this time. The solution is valid for all directions of the external field relative to the crystal axes. This makes the anisotropy of the upper critical field into an unambiguous test of unconventional superconductivity. Good agreement with experimental measurements on ${\mathrm{UPt}}_3$ is found.