Critical magnetic fields in a superconductor coupled to a superfluid

We study a superconductor that is coupled to a superfluid via density and derivative couplings. Starting from a Lagrangian for two complex scalar fields, we derive a temperature-dependent Ginzburg-Landau potential, which is then used to compute the phase diagram at nonzero temperature and external magnetic field. This includes the calculation of the critical magnetic fields for the transition to an array of magnetic flux tubes, based on an approximation for the interaction between the flux tubes. We find that the transition region between type-I and type-II superconductivity changes qualitatively due to the presence of the superfluid: the phase transitions at the upper and lower critical fields in the type-II regime become first order, opening the possibility of clustered flux tube phases. These flux tube clusters may be realized in the core of neutron stars, where superconducting protons are expected to be coupled to superfluid neutrons.