${}^{3}P_2$ superfluids are topological

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.

Figure 1

(a) GL phase diagram. (b) Gap and topological structures of nematic phases. The thick arrows represent the $\mathit{d}$ vectors and the inner (red-colored) sphere denotes the Fermi sphere. (c) Phase diagram under a magnetic field, obtained from the superfluid Fermi liquid theory. The UN phase is stabilized at $H=0$ for $T<T_{\mathrm{c}0}$. The thick (thin) curve is the first- (second-) order phase boundary. (d) $C\left(T\right)/C_{\mathrm{N}}\left(T\right)$ under fixed magnetic fields.

Figure 2

(a) Momentum-resolved surface density of states ${\mathcal{N}}_{\mathrm{s}}\left(\widehat{\mathit{k}},E\right)$ and (b) local magnetization for $\mathit{H}\perp \widehat{\mathit{z}}$ and $\mathit{H}\parallel \widehat{\mathit{z}}$, in the nematic phase with $\widehat{\mathit{u}}\perp \widehat{\mathit{z}}$.

Figure 3

(a) Configuration of Weyl points in the cyclic phase, where $q=\pm 1$ possess the monopole charge $q_{\mathrm{m}}=\pm 1$. (b) Momentum-resolved zero-energy density of states on the surface for the misorientation angle $\vartheta =2\pi /5$.