Emergence of the Geometric Contribution to the Superfluid Density in the Inner Crust of Neutron Stars

The geometric contribution to the superfluid density has been found to be of great importance in the inner crust of neutron stars. In this work we clarify how this contribution arises in the context of a band theory for neutrons. Specifically, we derive the dependence of the superfluid density on the magnitude of the pairing gap when the system has many bands cutting the Fermi energy, as it is the case for the neutrons in the inner crust. Also, in the perturbation theory framework, we find that it is essential to account for the corrections to the (Bogoliubov) quasi-particle states in order to get the geometric contribution. Accounting only for the corrections to the (Hartree-Fock) single-particle states leads to the conventional contribution only.