Flux tubes and the type-I/type-II transition in a superconductor coupled to a superfluid

We analyze magnetic-flux tubes at zero temperature in a superconductor that is coupled to a superfluid via both density and gradient (" entrainment") interactions. The example we have in mind is high-density nuclear matter, which is a proton superconductor and a neutron superfluid, but our treatment is general and simple, modeling the interactions as a Ginzburg-Landau effective theory with four-fermion couplings, including only s -wave pairing. We numerically solve the field equations for flux tubes with an arbitrary number of flux quanta and compare their energies. This allows us to map the type-I/type-II transition in the superconductor, which occurs at the conventional κ λ/ξ=1/2 if the condensates are uncoupled. We find that a density coupling between the condensates raises the critical κ and, for a sufficiently high neutron density, resolves the type-I/type-II transition line into an infinite number of bands corresponding to " type-II (n) " phases, in which n, the number of quanta in the favored flux tube, steps from 1 to infinity. For lower neutron density, the coupling creates spinodal regions around the type-I/type-II boundary, in which metastable flux configurations are possible. We find that a gradient coupling between the condensates lowers the critical κ and creates spinodal regions. These exotic phenomena may not occur in nuclear matter, which is thought to be deep in the type-II region but might be observed in condensed-matter systems.