Suppression of superfluid density in the superfluid-supersolid   transition
  We show that the rather unexpected pressure dependence of superfluid density observed near the superfluid-supersolid transition by Kim {\em et.al.}[M.H.W. Chan, {\em private communication}], can be understood if the transition from superfluid to supersolid state is a second order or weakly first order transition from the superfluid state to a super-CDW state with non-uniform Bose-condensation amplitude. The suppression of superfluid density is a direct consequence of softening of phonon mode at finite wave-vector $|\vec{Q}|\sim Q_0$ around the quantum phase transition. 
Tai Kai Ng