Collective modes in a superfluid neutron gas within the quasiparticle random-phase approximation

We study collective excitations in a superfluid neutron gas at zero temperature within the quasiparticle random-phase approximation. The particle-hole residual interaction is obtained from a Skyrme functional, while a separable interaction is used in the pairing channel, which gives a BCS gap that is very similar to the one obtained with a realistic nucleon-nucleon interaction. In accordance with the Goldstone theorem, we find an ungapped collective mode (analogous to the Bogoliubov-Anderson mode). At low momentum, its dispersion relation is approximately linear and its slope coincides with the hydrodynamic speed of sound calculated with the Skyrme equation of state. The response functions are compared with those obtained within the Landau approximation. We also compute the contribution of the collective mode to the specific heat of the neutron gas.

Figure 1

Value of the gap at the Fermi surface, ${\Delta }_{k_F}$, as function of the Fermi momentum $k_F$, obtained with the separable interaction (solid line) and with the $V_{\text{low-}k}$ interaction of [10] (dashes).

Figure 2

QRPA (solid lines) and RPA (dashes) response functions at density $\rho =0.04$ ${\mathrm{fm}}^{-3}$, as functions of the excitation energy $\omega$ for two different momentum transfers $q=k_F$ (a) and $2k_F$ (b).

Figure 3

QRPA response functions for $q/k_F=0.5,1,1.3$ (a) and $2,2.5,3$ (b) at $\rho =0.016$ ${\mathrm{fm}}^{-3}$ as functions of the excitation energy $\omega$. The arrows in the upper panel represent $\delta$-function peaks corresponding to the undamped collective modes. Their height is proportional to their strength, which corresponds to 71.3% $\left(q=0.5k_F\right)$, 25.2% $\left(q=k_F\right)$, and 9.5% $\left(q=1.3k_F\right)$ of the total strength of the response function. In the lower panel (b), the collective mode lies above the continuum threshold.

Figure 4

Dispersion relation ${\omega }_{\mathbf{q}}$ of the undamped (solid line) and damped (dashes) collective mode at $\rho =0.016$ (a) and $0.04$ ${\mathrm{fm}}^{-3}$ (b). At small $q$, it agrees with the linear dispersion relation $\omega =uq$ of hydrodynamic sound (dash-dotted line). At higher $q$, it approaches and finally crosses the pair-breaking threshold (dotted line).

Figure 5

Response functions obtained within the full QRPA (solid lines) and within the Landau approximation including only $F_0$ (dotted lines) or $F_0$ and $F_1$ (dashes) as functions of the excitation energy $\omega$ for neutron densities $\rho =0.016$ ${\mathrm{fm}}^{-3}$ (lower panels) and $0.04$ ${\mathrm{fm}}^{-3}$ (upper panels) and momenta $q=0.3$ ${\mathrm{fm}}^{-1}$ (left panels) and $1.5$ ${\mathrm{fm}}^{-1}$ (right panels). The arrows in the left panels indicate positions and strengths of the collective modes. In QRPA, the strengths of the collective modes correspond to 82% $(\rho =0.016$ ${\mathrm{fm}}^{-3})$ and 52% $(\rho =0.04$ ${\mathrm{fm}}^{-3})$ of the total strength of the response functions.

Figure 6

Heat capacity of a neutron gas with density $\rho =0.003$ (a) and $0.0184$ ${\mathrm{fm}}^{-3}$ (b), corresponding to total baryon densities in the neutron-star crust of ${\rho }_B\approx 0.00373$ and $0.0204$ ${\mathrm{fm}}^{-3}$, respectively: neutron quasiparticle contribution (dashes) and contribution of the collective mode calculated within QRPA (solid lines) and within the hydrodynamic approximation (dashed-dotted lines). For comparison, we also display the electron contribution (dotted lines) under the assumption of ${\mu }_e=36.2$ (a) and $50.1$ MeV (b), corresponding to electron densities ${\rho }_e=2.1\times {10}^{-4}$ and $5.5\times {10}^{-4}$ ${\mathrm{fm}}^{-3}$.