Chiral Phases of Superfluid $^3$He in an Anisotropic Medium

I report theoretical results for the phases of superfluid $^3$He infused into homogeneous uniaxial aerogel. Ginzburg-Landau (GL) theory for a class of equal-spin-pairing (ESP) states in a medium with uniaxial anisotropy is developed and used to analyze recent experiments on uniaxially strained aerogels. For $^3$He in an axially ``stretched'' aerogel GL theory predicts a transition from normal liquid into a \emph{chiral} ABM phase in which the chirality axis is aligned along the strain axis. This state is protected from random fluctuations in the anisotropy direction, has a positive NMR shift, a sharp NMR resonance line and is in quantitative agreement with NMR in the high-temperature ESP-1 phase of superfluid $^3$He in axially stretched aerogel. A second transition into a bi-axial phase is predicted to onset at a slightly lower temperature. This phase is an ESP state, breaks time-reversal symmetry, and is defined by an order parameter that spontaneously breaks axial rotation symmetry. The bi-axial phase has a continuous degeneracy associated with broken axial symmetry. Theoretical predictions for the NMR frequency shifts provide an identification of the ESP-2 phase as the bi-axial state, partially \emph{disordered} by random anisotropy (Larkin-Imry-Ma effect).