Phase structure of neutron $^{3}P_{2}$ superfluids in strong magnetic fields in neutron stars

We discuss the effect of a strong magnetic field on neutron $^{3}P_{2}$ superfluidity. Based on the attraction in the $^{3}P_{2}$ pair of two neutrons, we derive the Ginzburg-Landau equation in the path-integral formalism by adopting the bosonization technique and leave the next-to-leading order in the expansion of the magnetic field $B$. We determine the $(T,B)$ phase diagram with temperature $T$, comprising three phases: the uniaxial nematic (UN) phase for $B=0$, D$_{2}$-biaxial nematic (BN) and D$_{4}$-BN phases in finite $B$ and strong $B$ such as magnetars, respectively, where D$_{2}$ and D$_{4}$ are dihedral groups. We find that, compared with the leading order in the magnetic field known before, the region of the D$_{2}$-BN phase in the $(T,B)$ plane is extended by the effect of the next-to-leading-order terms of the magnetic field. We also present the thermodynamic properties, such as heat capacities and spin susceptibility, and find that the spin susceptibility exhibits anisotropies in the UN, D$_{2}$-BN, and D$_{4}$-BN phases. This information will be useful to understand the internal structures of magnetars.