Strong-coupling and the stability of crystalline order in superfluid $^3$He films*

In a film of thickness $D$, weak-coupling theory for p-wave, spin-triplet pairing predicts a ``stripe'' phase that spontaneously breaks translational symmetry in the plane of the film.\footnote{Vorontsov \& Sauls. \emph{Phys. Rev. Lett.} 98, 2007.} NMR on superfluid $^3$He confined in a slab has so far failed to detect any signature of the stripe phase, and the A-B transition is observed at lower temperatures than predicted by weak-coupling theory.\footnote{Levitin et al. \emph{Science} 340, 2013.} We report calculations of the phase diagram for $^3$He films based on Ginzburg-Landau (GL) theory that includes strong-coupling effects via experimentally estimated $\beta$ parameters.\footnote{Choi et al. \emph{Phys. Rev. B} 75, 2007.} At low pressures GL theory predicts the A-stripe phase transition, for small $D$, to be significantly suppressed compared to weak-coupling. For large $D$, the stripe transition is eliminated in favor of an A-B transition at lower temperatures than in weak-coupling. At higher pressures the stripe phase is predicted to be stable only at very low temperatures, outside the expected applicability of the strong-coupling GL theory. Our results suggest that the discrepancy between experiment and weak-coupling theory likely results from strong-coupling effects.