Reexamining Ginzburg-Landau theory for neutron ${}^{3}P_2$ superfluidity in neutron stars

The Ginzburg-Landau (GL) effective theory is a useful tool to study a superconductivity or superfluidity near the critical temperature, and usually the expansion up to the fourth-order in terms of order parameters is sufficient for the description of the second-order phase transition. In this paper, we discuss the GL equation for the neutron ${}^{3}P_2$ superfluidity relevant for interior of neutron stars. We derive the GL expansion up to the eighth order in the condensates and find that this order is necessary for the system to have the unique ground state, unlike the ordinary cases. Starting from the $LS$ potential, which provides the dominant attraction between two neutrons at the high density, we derive the GL equation in the path-integral formalism, where the auxiliary field method and the Nambu-Gor'kov representation are used. We present the detailed description for the trace calculation necessary in the derivation of the GL equation. As numerical results, we show the phase diagram of the neutron ${}^{3}P_2$ superfluidity on the plane spanned by the temperature and magnetic field, and we find that the eighth-order terms lead to a first-order phase transition, whose existence was predicted in the Bogoliubov-de Gennes equation but has not been found thus far within the framework of the GL expansion up to the sixth order. The first-order phase transition will affect the interior structures inside the neutron stars.

Figure 1

The plots of the GL free energy density $\mathrm{\Delta }f=f-f_0$ as functions of $A_0$ for $t=0.9$. (a) $b=0$ ($r=-1/2$) and (b) $b=1$ ($r=-1$). The dotted lines are the results in the previous study [setting ${\delta }^{\left(0\right)}=0$ in Eq. (30)], and the solid lines are the results up to $\mathcal{O}\left(A^8\right)$ in the present study (setting ${\delta }^{\left(0\right)}\ne 0$). The magenta points are the minimum points with respect to $A_0$.

Figure 2

The phase diagram for $A_0$ and $r$ on the plane spanned by the temperature ($t=T/T_{c0}$) and the magnetic field ($b={\gamma }_{n}B/\left[(1+F_0^a)T_{c0}\right]$). The two left panels (a, c) are the results in the previous study [setting ${\delta }^{\left(0\right)}=0$ in Eq. (30)] [73], and the two right panels (b, d) are the results up to $\mathcal{O}\left(A^8\right)$ in the present study [setting ${\delta }^{\left(0\right)}\ne 0$ in Eq. (30)]. In the bottom panels, we show the value of $r$ representing the phases, the UN phase ($r=-1/2$), the ${\mathrm{D}}_2-\mathrm{BN}$ phase ($-1<r<-1/2$), and the ${\mathrm{D}}_4-\mathrm{BN}$ phase ($r=-1$). The cyan lines in the right panels indicate the first-order phase transition.

Figure 3

The plots of $A_0$ and $r$ as functions of $t$ for $b=0$. The red line indicates $A_0\ge 0$, and the blue line indicates $-1\le r\le -1/2$. For $b=0, A_0$ and $r$ are discontinuous at the first-order phase transition at $t=0.772$.