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 Poincaré-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 .