Symmetry and topology of the boundary of neutron $^{3}P_{2}$ superfluids in neutron stars: boojums as surface topological defects. (arXiv:1905.13666v1 [nucl-th])
<a href="http://arxiv.org/find/nucl-th/1/au:+Yasui_S/0/1/0/all/0/1">Shigehiro Yasui</a>, <a href="http://arxiv.org/find/nucl-th/1/au:+Chatterjee_C/0/1/0/all/0/1">Chandrasekhar Chatterjee</a>, <a href="http://arxiv.org/find/nucl-th/1/au:+Nitta_M/0/1/0/all/0/1">Muneto Nitta</a>
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 facing to
the boundary at the surface sphere. 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/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 (OPM) 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
anti-vortices depend on the bulk phases, the difference between them are
topologically invariant (the Euler number $chi=2$) irrespective to 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.
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 facing to
the boundary at the surface sphere. 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/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 (OPM) 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
anti-vortices depend on the bulk phases, the difference between them are
topologically invariant (the Euler number $chi=2$) irrespective to 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.
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