Elastic properties and thermodynamic anomalies of supersolids

We study a supersolid in the context of a Gross-Pitaevskii theory with a non-local effective potential. We employ a homogenisation technique which allows us to calculate the elastic moduli, supersolid fraction and other state variables of the system. Our methodology is verified against numerical simulations of elastic deformations. We can also verify that the long-wavelength Goldstone modes that emerge from this technique agree with Bogoliubov theory. We find a thermodynamic anomaly that the supersolid does not obey the thermodynamic relation $\partial P / \partial V \bigr|_N = - n \, \partial P / \partial N \bigr|_V$, which we claim is a feature unique to supersolids.