Collective excitations of a quantized vortex in ${}^{3}P_2$ superfluids in neutron stars

We discuss collective excitations (both fundamental and solitonic excitations) of quantized superfluid vortices in neutron ${}^{3}P_2$ superfluids, which likely exist in high density neutron matter such as neutron stars. Besides the well-known Kelvin modes (translational zero modes), we find a gapful mode whose low-energy description takes the simple form of a double sine-Gordon model. The associated kink solution and its effects on spontaneous magnetization inside the vortex core are analyzed in detail.

Figure 1

We display the profile functions $f_1, f_2$, and $g$ for a vortex configuration of winding number 1 as a function of the distance $\tilde{\rho }$ from the vortex center. Here, the order parameter is expressed in the cylindrical basis ($n=1$) and the vortex solution of lowest energy is found for the boundary value $\eta =-0.557$. Numerical results are shown for the case $m=0$.

Figure 2

Magnetization $\sigma \left(\tilde{\rho }\right)$ for $\delta =0$ plotted as a function of the rescaled distance $\tilde{\rho }$ from the vortex center.

Figure 3

We display the effective free energy potential $F_{\text{eff}}$ (3.3) as a function of the $\delta$ parameter. We evaluate the free energy for the $\lambda$ coefficients listed in Eq. (3.5). The GL parameters are given by $\alpha =-0.1$ and $\beta =2.57664\text{(MeV}{\text{fm)}}^{-2}$. The coherence length of the system is set to $30.3634\text{fm}$. Note that the energy values are given in units of MeV ${\text{fm}}^{-1}$.

Figure 4

The double sine-Gordon kink solution $\delta \left(\zeta \right)$ (4.7) shown for $\mathrm{\Lambda }=0.813959$.

Figure 5

The magnetization $\left(\frac{2m\left(\zeta \right)}{|{\gamma }_n|\hslash }=-\sigma \left(\zeta \right)\right)$ plotted as a function of the rescaled $z$ coordinate $\zeta$.

Figure 6

A schematic figure of a magnetic field configuration to illustrate the flipped magnetic moment and the leakage. The dotted red arrows and the solid blue arrows denote the magnetic field $\mathit{H}$ and magnetic moment $\mathit{M}$, respectively.