Patterns and supersolids

In the frame of the Gross-Pitaevskii equation with a non-local interaction term we study the crystallization induced by the non-local interaction and its connection with the theory of supersolids. The crystallization is well understood as the search of a ground-state of a certain energy functional, and uses the techniques of pattern-formation in the weak (but finite) amplitude limit. In two space dimensions an hexagonal pattern is displayed, however, in three space dimensions density modulation displays an hexagonal-close-packing or hcp structure.We derive, using the technique of homogenization, an effective Lagrangian that provides the long-wave slow dynamics for the local density variations, the global wavefunction phase, and for the crystal deformation. As a classical crystal, our supersolid displays shear and compression waves for the elastic deformations, however the later are coupled with the wave function phase and the local density. Finally the system presents the quite remarkable property of non-classical rotational inertia (NCRI) under rotation. Indeed under an uniform slow rotation the effective moment of inertia is less than the one of a classical rigid body.