Resonant absorption at the vortex-core states ind-wave superconductors

We predict a resonant microwave absorption on collective vortex modes in a superclean $d$-wave superconductor at low temperatures. Energies of the collective modes are multiples of the distance between the exact quantum levels of bound states in the vortex core at lower temperatures and involve delocalized states for higher temperatures. The characteristic resonant frequency is larger than the cyclotron frequency ${\omega }_c$ but lower than the Caroli-de Gennes-Matricon minigap ${\Delta }^2{/E}_F;$ it has a $\sqrt{H}$ dependence on the magnetic field and decreases down to ${\omega }_c$ with increasing temperature. We calculate the vortex mass as a response to a slow acceleration. This mass is equal to the mass of electrons inside the vortex core region with the area of the order of ${\xi }^2\sqrt{H_{c2\mathrm{}}/H}$; it increases with temperature and diverges for $\overrightarrow{H}0$. We discuss the universal flux-flow regime predicted by Kopnin and Volovik [Phys. Rev. Lett. 79, 1377 (1997)] and show that it exists in a broader temperature range than has been originally found.