  In this contribution we first review a method for obtaining the collective
modes of pair-correlated neutron matter as found in a neutron star inner crust.
We discuss two classes of modes corresponding to density and spin perturbations
with energy spectra $\omega = \omega_0 + \alpha q^2$, where $\omega_0 =
2\Delta$ is the threshold frequency and $\Delta$ is the gap in the neutron
fluid spectrum. For characteristic values of Landau parameters in neutron star
crusts the exitonic density modes have $\alpha < 0$, therefore they live below
the pair-breaking continuum $\omega_0$ and are undamped. For spin diffusive
modes $\alpha > 0$ and they exist above $\omega_0$ which implies that these
modes are damped. As an application of these findings we compute the thermal
conductivity due to spin diffusive modes and show that it scales as $T^{1/2}
\exp(-2\omega_0/T)$ in the case where their two-by-two scattering cross-section
is weakly dependent on temperature.
