Superfluid fraction tensor of a two-dimensional supersolid

We investigate the superfluid fraction of crystalline stationary states within the framework of mean-field Gross-Pitaevskii theory. Our primary focus is on a two-dimensional Bose-Einstein condensate with a non-local soft-core interaction, where the superfluid fraction is described by a rank-2 tensor. We then calculate the superfluid fraction tensor for crystalline states exhibiting triangular, square, and stripe geometries across a broad range of interaction parameters. Factors leading to an anisotropic superfluid fraction tensor are also considered. We also refine the Leggett bounds for the superfluid fraction of the 2D system. We systematically compare these bounds to our full numerical results, and other results in the literature. This work is of direct relevance to other supersolid systems of current interest, such as supersolids produced using dipolar Bose-Einstein condensates.