Nucleon superfluids which are realized in neutron star interior and symmetric nuclear matter are studied with use of realistic nuclear forces, in the density domain from the subnuclear region to about 3ρ_0 (ρ_0 being the nuclear density). It is shown that characteristic aspects of nuclear forces manifest themselves in the appearance of several kinds of nucleon superfluids, which strongly depends on the density ρ. In this chapter emphasis is put on the pairing correlations where strong noncentral (tensor and spin-orbit) forces play important roles. A theoretical framework applicable to the nonzero angular-momentum pairing including the coupling due to tensor force is given by extending the usual BCS-Bogoliubov theory for the ^1S_0 pairing (the zero angular-momentum one). This formulation has been applied to the ^3P_2+^3F_2 pairing in neutron matter (the dominant component of neutron stars) and the ^3S_1+^3D_1 pairing in symmetric nuclear matter. In the former case, although spin-orbit force mainly contributes to the ^3P_2 attraction, the tensor coupling with the ^3F_2 component assists to realize the ^3P_2 superfluid. In the latter case, the tensor coupling to the ^3D_1 component plays a vital role to realize the ^3S_1 superfluid with a large energy gap. Results of the energy gaps calculated for such nonzero angular-momentum pairings, as well as those for the ^1S_0 pairing, are shown. We have found the realization of the following nucleon superfluids; the neutron ^3P_2 superfluid and the proton ^1S_0 one in the fluid core of neutron stars at ρ≃(0.7∼3)ρ_0, the neutron ^1S_0 superfluid in the inner crust of neutron stars at ρ≃(10^−3∼0.5)ρ_0, and the ^3S_1 superfluid in symmetric nuclear matter at a wide range of ρ including ρ_0, contrary to the ^1S_0 one realized at ρ≲ρ_0/2. The properties of these superfluids and their implications are also discussed.

Superfluidity in neutron star matter and symmetric nuclear matter

We discuss quantized vortices in neutron $^{3}P_{2}$ superfluids, which are believed to realize in high density neutron matter such as neutron stars. By using the Ginzburg-Landau free energy for $^{3}P_{2}$ superfluids, we determine the ground state in the absence and presence of the external magnetic field, and numerically construct $^{3}P_{2}$ quantized vortices in the absence and presence of the external magnetic field along the vortex axis (poloidal) or angular direction (toroidal). We find in certain situations the spontaneous magnetization of the vortex core, whose typical magnitude is about ${10}^{7--8}$ G, but the net magnetic field in a neutron star is negligible because of the ratio of the vortex core size $\ensuremath{\sim}10\phantom{\rule{0.28em}{0ex}}\mathrm{fm}$ and the intervortex distance $\ensuremath{\sim}{10}^{\ensuremath{-}6}\phantom{\rule{0.28em}{0ex}}\mathrm{m}$ in a vortex lattice.

Magnetic properties of quantized vortices in neutron P 2 3 superfluids in neutron stars

We clarify the topology of $^{3}P_{2}$ superfluids which are expected to be realized in the inner cores of neutron stars and cubic odd-parity superconductors. $^{3}P_{2}$ phases include uniaxial/biaxial nematic phases and nonunitary ferromagnetic and cyclic phases. Here, we show that all the phases are accompanied by different types of topologically protected gapless fermions: surface Majorana fermions in nematic phases and a quartet of (single) itinerant Majorana fermions in the cyclic (ferromagnetic) phase. Using the superfluid Fermi liquid theory, we also demonstrate that dihedral-two and -four biaxial nematic phases are thermodynamically favored in the weak coupling limit under a magnetic field. It is shown that the tricritical point exists on the phase boundary between these two phases and may be realized in the core of realistic magnetars. We unveil the intertwining of symmetry and topology behind the mass acquisition of surface Majorana fermions in nematic phases.

P 2 3 superfluids are topological

We study the topology and symmetry of surface Majorana arcs in superconductors with nonunitary ``cyclic'' pairing. Cyclic $p$-wave pairing may be realized in a cubic or tetrahedral crystal, while it is a candidate for the interior $^{3}\mathrm{P}_{2}$ superfluids of neutron stars. The cyclic state is an admixture of full gap and nodal gap with eight Weyl points and the low-energy physics is governed by itinerant Majorana fermions. We here show the evolution of surface states from Majorana cone to Majorana arcs under rotation of surface orientation. The Majorana cone is protected solely by an accidental spin rotation symmetry and fragile against spin-orbit coupling, while the arcs are attributed to two topological invariants: the first Chern number and one-dimensional winding number. Lastly, we discuss how topologically protected surface states inherent to the nonunitary cyclic pairing can be captured from surface probes in candidate compounds, such as ${\mathrm{U}}_{1\ensuremath{-}x}{\mathrm{Th}}_{x}{\mathrm{Be}}_{13}$. We examine tunneling conductance spectra for two competitive scenarios in ${\mathrm{U}}_{1\ensuremath{-}x}{\mathrm{Th}}_{x}{\mathrm{Be}}_{13}$---the degenerate ${E}_{u}$ scenario and the accidental scenario.

Topology and symmetry of surface Majorana arcs in cyclic superconductors

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 $B=0, {\mathrm{D}}_{2}$-biaxial nematic (BN) and ${\mathrm{D}}_{4}$-BN phases in finite $B$ and strong $B$ such as magnetars, respectively, where ${\mathrm{D}}_{2}$ and ${\mathrm{D}}_{4}$ are dihedral groups. We find that, compared with the leading order in the magnetic field known before, the region of the ${\mathrm{D}}_{2}$-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, ${\mathrm{D}}_{2}$-BN, and ${\mathrm{D}}_{4}$-BN phases. This information will be useful to understand the internal structure of magnetars.

Phase structure of neutron P 2 3 superfluids in strong magnetic fields in neutron stars

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The Ginzburg-Landau equation for 3P2 pairing Superfluidity in neutron stars

Hydrodynamic responses of superfluid 3HeA in a cylinder to time dependent rotational perturbations are studied. The characteristic time involved in deformation of the MerminHo structure is shown to be 100R2 ( Δ kT c ), where R is the radius of the cylinder. Possibilities of observing the responses are briefly discussed.

Toshimitsu Fujita

Papers

The Ginzburg-Landau equation for 3P2 pairing Superfluidity in neutron stars

Response of 3He in Mermin-Ho texture to rotational perturbation