Phase structure of neutron ${}^{3}P_2$ superfluids in strong magnetic fields in neutron stars

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.

Figure 1

The phase diagram for $A_0$ and $r$ on the $T-B$ plane. The two left panels (upper left and bottom left) are the results up to $O\left(B^2{A}^2\right)$ in Ref. [64] [setting ${\beta }^{\left(4\right)}={\gamma }^{\left(2\right)}=0$ in Eq. (13)], and the two right panels (upper right and bottom right) are the results up to $O\left(B^4{A}^2\right)+O\left(B^2{A}^4\right)$ in the present study. In the bottom panels, the phases are in the UN ($r=-1/2$), ${\mathrm{D}}_2$-BN ($-1<r<-1/2$), and ${\mathrm{D}}_4$-BN phases ($r=-1$).

Figure 2

The heat capacity $C(T,B)$ and the magnetic susceptibility $\chi (T,B)={\chi }_2(T,B)$ along the direction parallel to the magnetic field. They are normalized by $C_N(T,B)$ and ${\chi }_N(T,B)$, respectively.

Figure 3

Spin susceptibilities ${\chi }_i(T,B)$ ($i=1,2,3$) for various magnitudes of magnetic field. They are normalized by ${\chi }_N(T,B)$.