Mixing of charged and neutral Bose condensates at nonzero temperature and magnetic field

It is expected that in the interior of compact stars a proton superconductor coexists with and couples to a neutron superfluid. Starting from a field-theoretical model for two complex scalar fields – one of which is electrically charged – we derive a GinzburgLandau potential which includes entrainment between the two fluids and temperature effects from thermal excitations of the two scalar fields and the gauge field. The GinzburgLandau description is then used for an analysis of the phase structure in the presence of an external magnetic field. In particular, we study the effect of the superfluid on the flux tube phase by computing the various critical magnetic fields and deriving an approximation for the flux tube interaction. As a result, we point out differences to the naive expectations from an isolated superconductor, for instance the existence of a first-order flux tube onset, resulting in a more complicated phase structure in the region between type-I and type-II superconductivity.