Disorder-induced Missing Moment of Inertia in Solid $^4$He

A microscopic model for the experimentally observed missing moment of inertia in the torsional oscillator experiments on partially-annealed solid $^4$He is proposed. We argue that an ordered array of $^4$He atoms is a Mott insulator. Disorder destroys the Mott state producing localized states in the gap which beyond a critical value of the disorder induce a superfluid state. Depending on the magnitude of the disorder, we find that the destruction of the Mott state takes place for $d\le 3$ either through a Bose glass phase (strong disorder and weak disorder) or through a direct transition to a superfluid (intermediate disorder).The critical value of the disorder that separates three region of disorder is shown to be a function of the boson filling, interaction and the momentum cut off. We apply our work to the experimentally observed enhancement $^3 $He impurities has on the onset temperature for the missing moment of inertia. We find quantitative agreement with experimental trends.