Supersolid State of Matter

We prove that the necessary condition for a solid to be also a superfluid is to have zero-point vacancies, or interstitial atoms, or both, as an integral part of the ground state. As a consequence, in the absence of symmetry between vacancies and interstitials, superfluidity has a zero probability to occur in commensurate solids which break continuous translation symmetry. We discuss recent ${}^4\mathrm{He}$ experiments by Kim and Chan in the context of this theorem, question its bulk supersolid interpretation, and offer an alternative explanation in terms of superfluid interfaces.