Topological defects at the boundary of neutron $^{3}P_{2}$ superfluids in neutron stars

We study surface effects of neutron $^{3}P_{2}$ superfluids in neutron stars. $^{3}P_{2}$ superfluids are in uniaxial nematic (UN), D$_{2}$ biaxial nematic (BN), or D$_{4}$ BN phase, depending on the strength of magnetic fields from small to large. We suppose a neutron $^{3}P_{2}$ superfluid in a ball with a spherical boundary. Adopting a suitable boundary condition for $^{3}P_{2}$ condensates, we solve the Ginzburg-Landau equation to find several surface properties for the neutron $^{3}P_{2}$ superfluid. First, the phase on the surface can be different from that of the bulk, and symmetry restoration or breaking occurs in general on the surface. Second, the distribution of the surface energy density has an anisotropy depending on the polar angle in the sphere, which may lead to the deformation of the geometrical shape of the surface. Third, the order parameter manifold induced on the surface, which is described by two-dimensional vector fields induced on the surface from the condensates, allows topological defects (vortices) on the surface, and there must exist such defects even in the ground state thanks to the Poincar\'{e}-Hopf theorem: although the numbers of the vortices and antivortices depend on the bulk phases, the difference between them is topologically invariant (the Euler number $\chi=2$) irrespective of the bulk phases. These vortices, which are not extended to the bulk, are called boojums in the context of liquid crystals and helium-3 superfluids. The surface properties of the neutron $^{3}P_{2}$ superfluid found in this paper may provide us useful information to study neutron stars.