Coexistence phase of S0 1 and P2 3 superfluids in neutron stars

In neutron star matter, there exist S01 superfluids in lower density in the crust while P23 superfluids are believed to exist at higher density deep inside the core. In the latter, depending on the temperature and magnetic field, either the uniaxial nematic phase, the D2-biaxial nematic phase, or the D4-biaxial nematic (D4-BN) phase appears. In this paper, we discuss a mixture of the S01 and P23 superfluids and find their coexistence. Adopting the loop expansion and the weak-coupling approximation for the interaction between two neutrons, we obtain the Ginzburg-Landau (GL) free energy in which both of the S01 and P23 condensates are taken into account by including the coupling terms between them. We analyze the GL free energy and obtain the phase diagram for the temperature and magnetic field. We find that the S01 superfluid excludes the P23 superfluid completely in the absence of magnetic field, they can coexist for weak magnetic fields, and the S01 superfluid is expelled by the P23 superfluid at strong magnetic fields, thereby proving the robustness of P23 superfluid against the magnetic field. We further show that the D4-BN phase covers the whole region of the P23 superfluidity as a result of the coupling term, in contrast to the case of a pure P23 superfluid studied before in which the D4-BN phase is realized only under strong magnetic fields. Thus, the D4-BN phase is topologically the most interesting phase, e.g., admitting half-quantized non-Abelian vortices relevant not only in magnetars but also in ordinary neutron stars.