  We have studied systematically various microscopic properties of a quantum
vortex in neutron matter at different temperatures and densities corresponding
to different layers of the inner crust of a neutron star. To this end, we have
carried out fully self-consistent 3D Hartree-Fock-Bogoliubov calculations,
using one of the latest nuclear energy-density functionals from the
Brussels-Montreal family, which has been developed specifically for
applications to neutron superfluidity in neutron-star crusts. By analyzing the
flow around the vortex, we have determined the effective radius relevant for
the vortex filament model. We have also calculated the specific heat in the
presence of the quantum vortex and have shown that it is substantially larger
than for a uniform system at low temperatures. We have identified the low
temperature limit of the specific heat as being determined by Andreev states
inside the vortex core. The typical energy scale associated with Andreev states
is defined by the minigap, which we have extracted for various neutron-matter
densities. Our results suggest that vortices may be spin-polarized in the crust
of magnetars. Finally, we have discussed how to evaluate the specific heat for
an arbitrary surface density of vortices, taking into account both the
contributions from the vortex core states and from the hydrodynamic flow.
