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In neutron star matter, there exist $^{1}S_{0}$ superfluids in lower densityin the crust while $^{3}P_{2}$ superfluids are believed to exist at higherdensity deep inside the core. In the latter, depending on the temperature andmagnetic field, either the uniaxial nematic phase, the D$_{2}$-biaxial nematicphase, or the D$_{4}$-biaxial nematic phase appears. In this paper, we discussa mixture of the $^{1}S_{0}$ and $^{3}P_{2}$ superfluids and find theircoexistence. Adopting the loop expansion and the weak-coupling approximationfor the interaction between two neutrons, we obtain the Ginzburg-Landau (GL)free energy in which both of the $^{1}S_{0}$ and $^{3}P_{2}$ condensates aretaken into account by including the coupling terms between them. We analyze theGL free energy and obtain the phase diagram for the temperature and magneticfield. We find that the $^{1}S_{0}$ superfluid excludes the $^{3}P_{2}$superfluid completely in the absence of magnetic field, they can coexist forweak magnetic fields, and the $^{1}S_{0}$ superfluid is expelled by the$^{3}P_{2}$ superfluid at strong magnetic fields, thereby proving therobustness of $^{3}P_{2}$ superfluid against the magnetic field. We furthershow that the D$_{4}$-BN phase covers the whole region of the $^{3}P_{2}$superfluidity as a result of the coupling term, in contrast to the case of apure $^{3}P_{2}$ superfluid studied before in which the D$_{4}$-BN phase isrealized only under strong magnetic fields. Thus, the D$_{4}$-BN phase istopologically the most interesting phase, e.g., admitting half-quantizednon-Abelian vortices relevant not only in magnetars but also in ordinaryneutron stars.

Coexistence phase of $^{1}S_{0}$ and $^{3}P_{2}$ superfluids in neutron stars