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Recent advances in experiment and theory suggest that superfluid $^3$He underplanar confinement may form a pair density wave (PDW) whereby superfluid andcrystalline orders coexist. While a natural candidate for this phase is aunidirectional stripe phase predicted by Vorontsov and Sauls in 2007, recentnuclear magnetic resonance measurements of the superfluid order parameterrather suggest a two-dimensional PDW with noncollinear wavevectors, of possiblysquare or hexagonal symmetry. In this Letter, we present a general mechanism bywhich a PDW with the symmetry of a triangular lattice can be stabilized, basedon a superfluid generalization of Landau's theory of the liquid-solidtransition. A soft-mode instability at finite wavevector within thetranslationally invariant planar-distorted B phase triggers a transition fromuniform superfluid to PDW that is first order due to a cubic term generallypresent in the PDW free-energy functional. This cubic term also lifts thedegeneracy of possible PDW states in favor of those for which wavevectors addto zero in triangles, which in two dimensions uniquely selects the triangularlattice.

Triangular pair density wave in confined superfluid $^3$He