Supersolid and the non-uniform superfluid

We construct a model of non-uniform superfluid having a spatially modulated order parameter that makes it kinematically an x-ray solid, but one with an associated superfluid flow. In the mean field approximation, such a supersolid is found to be energetically stabilized by self-interaction favouring a density wave. Intrinsic to this state is a non-classical translational inertia which we derive for the case of potential flow. Connection to the non-classical rotational inertia observed in recent experiments on solid helium-4 is discussed. Recent experiments of Kim and Chan [1,2] on solid helium-4 at very low temperatures have strikingly revealed a non-classical rotational inertia that seems intrinsic to it, much as is the case for superfluid He II. Such a supersolid was indeed predicted much earlier on theoretical grounds as a plausible concomitant of a quantum crystal with delocalized defects, or of a Bose-Einstein condensate [3-5], and had motivated years of research [6]. The non-classical inertial effect was, however, estimated to be very small, and direct tests were therefore suggested [5]. Thus, the question “can a solid be superfluid?”, raised some 35 years ago [5] has now been finally answered in the affirmative. In this work we explicitly construct a complex order-parameter state modeling a supersolid, and demonstrate analytically its necessarily non-classical inertia − the signature of a supersolid. In order to motivate an order-parameter approach to the supersolid, let us recall that geometrically a solid is a periodic spatial modulation of matter density. Such a crystalline solid structure will be revealed kinematically in its characteristic x-ray diffraction − we may call it an x-ray solid [3]. Such a kinematic description, however must be supplemented by the energetics of its stability against deformation. We will show below that both these conditions are intrinsically realized in our order-parameter description of the supersolid giving non-zero superfluid flow. There, we will explicitly consider a potential flow and show that the translational inertia of the supersolid is smaller than its literal mass. We will also comment on its relation to the diminished rotational inertia observed in the