Properties of a quantum vortex in neutron matter at finite temperatures

We have studied systematically microscopic properties of a quantum vortex in neutron matter at finite temperatures and densities corresponding to different layers of the inner crust of a neutron star. To this end and in preparation of future simulations of the vortex dynamics, we have carried out fully self-consistent three-dimensional (3D) Hartree-Fock-Bogoliubov calculations, using one of the latest nuclear energy-density functionals from the Brussels-Montreal family, which has been developed specifically for applications to neutron superfluidity in neutron-star crusts. By analyzing the flow around the vortex, we have determined the effective radius relevant for the vortex filament model. We have also calculated the specific heat in the presence of the quantum vortex and have shown that it is substantially larger than for a uniform system at low temperatures. The low temperature limit of the specific heat has been identified as being determined by Andreev states inside the vortex core. We have shown that the specific heat in this limit does not scale linearly with temperature. The typical energy scale associated with Andreev states is defined by the minigap, which we have extracted for various neutron-matter densities. Our results suggest that vortices may be spin-polarized in the crust of magnetars. Finally, we have obtained a lower bound for the specific heat of a collection of vortices with given surface density, taking into account both the contributions from the vortex core states and from the hydrodynamic flow.

Figure 1

${}^{1}S_0$ pairing gaps in uniform neutron matter as a function of density $\rho$, as calculated in Ref. [40] including both medium polarization effects and self-energy corrections. The six red points indicate the densities considered in this article.

Figure 2

Relative deviations $\delta \mathrm{\Delta }/\mathrm{\Delta }$ between the calculated ${}^{1}S_0$ pairing gap in uniform neutron matter and the reference one as a function of the cutoff energy $E_c$ for different densities $\rho$ in ${\mathrm{fm}}^{-3}$. As expected from the weak-coupling approximation $\left|\mathrm{\Delta }\right|\ll \mu$ underlying the adopted pairing functional, the error is the highest for the lowest density. In the vicinity of the maximum pairing gap, the overall precision of the self-consistent calculations lies within $5\%$.

Figure 3

Three-dimensional view of a vortex in neutron matter at density $\rho =0.0189{\mathrm{fm}}^{-3}$. The red translucent contour marks the region where the modulus of the pairing field $\left|\mathrm{\Delta }\right(\mathit{r}\left)\right|$ corresponds to 15% of the bulk pairing gap ${\mathrm{\Delta }}_{\infty }=1.55\mathrm{MeV}$. The line at the center shows the vortex core, while the outer tube is connected to boundary effects and roughly delimits the range of the external potential. Value of $\left|\mathrm{\Delta }\right(\mathit{r}\left)\right|$ are plotted at the bottom of the cube (the system is translationally invariant along the $z$ axis). Arrows at mid-height show the vector current $\mathit{j}\left(\mathit{r}\right)$.

Figure 4

Section through the center of a vortex core inside neutron matter. The upper figure shows the density profile (a), and the lower figure the pairing field (b). The neutron density and the modulus of the pairing field far from the vortex are ${\rho }_{\infty }=0.0059{\mathrm{fm}}^{-3}$ and ${\mathrm{\Delta }}_{\infty }=1.334$ MeV, respectively. The two length scales of the system are depicted in the plots: the coherence length $\xi$ and the inverse of the Fermi wave vector $k_{\mathrm{F}}^{-1}$.

Figure 5

(a) Size of the vortex core $R_{\mathrm{core}}$ in neutron matter as a function of temperature $T$ for different densities $\rho$. The size of the vortex increases with temperature and diverges at the critical point. (b) Tension per unit length of the vortex line as a function of temperature $T$ for different densities $\rho$. The tension grows with density and vanishes at the critical point.

Figure 6

(a) Norm of the velocity field $\mathit{v}\left(\mathit{r}\right)$ (in units of ${10}^{-2}c$), and (b) of the current $\mathit{j}\left(\mathit{r}\right)$ (in units of ${10}^{-5}c{\mathrm{fm}}^{-3}$) for the bulk neutron density $\rho =0.0059{\mathrm{fm}}^{-3}$ and the corresponding bulk pairing gap ${\mathrm{\Delta }}_{\infty }=1.334$ MeV. Three circles in the middle have radii $k_F^{-1},\xi ,R_{\mathrm{VFM}}$, respectively. The fourth circle near the boundary, with the radius $R_{\max }$, shows where the the boundary effects becomes non-negligible. (c) Cross section through the center of the vortex core. The orange solid curve denotes the velocity $\mathit{v}\left(\mathit{r}\right)$ associated with the mass current given by Eq. (20), while the purple dashed line depicts ${\mathit{v}}_{\mathrm{sf}}\left(\mathit{r}\right)$. While the former has a finite value in the core, the latter diverges. The green area shows the region where the difference between the two velocities is smaller than $10\%$. The radius $R_{\mathrm{VFM}}$ is defined as the distance from the vortex core to the point where the curves start to match.

Figure 7

(a) Superfluid fraction $\eta =v\left(\mathit{r}\right)/v_{\mathrm{sf}}\left(\mathit{r}\right)$ for different densities $\rho$ in the zero temperature limit. Far from the vortex core it saturates to 1, while it drops to zero in the core. (b) Density of the system relative to the bulk density ${\rho }_{\infty }$ far away from the vortex core. Note that the vertical axis does not start with 0, i.e., the core is filled with matter.

Figure 8

Distribution of angular momenta $m\hslash$ (along the vortex axis) of quasiparticle states of a given quasienergy $E_n$ [see Eq. (6)] in units of the Fermi energy ${\varepsilon }_{\mathrm{F}}={\hslash }^2{k}_F^2/\left(2M\right)$. The finite density in the vortex core allows for the existence of in-gap states with energies within the pairing gap $\mathrm{\Delta }$. The lowest energy state $E_{\mathrm{mg}}$ sets an important energy scale in the system. The data correspond to the vortex simulation for the bulk density ${\rho }_{\infty }=0.00590{\mathrm{fm}}^{-3}$.

Figure 9

Specific heat per unit of vortex length $C_V/L$ in neutron matter for different densities $\rho$ as a function of the temperature $T$. The vortex and uniform solutions are denoted by points and lines, respectively. (a) The specific heat of the system with a vortex is systematically higher than that of the corresponding uniform system. Note that, for the lowest density considered, the results are shown until the pairing gap dropped by 25% due to the temperature. For the remaining curves it corresponds to gaps $\approx 1$ MeV. (b) Specific heat in the vicinity of $T=0$ in logarithmic scale.

Figure 10

Ratio of the specific heat difference per unit length $\mathrm{\Delta }{C}_V/L$ with and without a vortex to the specific heat of the uniform system $C_V^{\text{uniform}}$ for various neutron densities $\rho$ as a function of temperature $T$. The thick solid line stands out from the other due to much weaker pairing.