Dynamic vortex mass in clean Fermi superfluids and superconductors

The vortex mass in superfluids and superconductors is a long standing problem in vortex physics and remains to be an issue of controversies. There are different approaches to its definition. One approach consists in calculating the vortex free energy increase due to the vortex velocity [1]. First used by Suhl [2] this approach yields the mass of the order of one quasiparticle mass (electron, in case of superconductor) per atomic layer (see Ref. [3] for an extensive review). Another approach is based on finding the force necessary to support an unsteady vortex motion. Identifying then the contribution to the force proportional to the vortex acceleration, one defines the vortex mass as a coefficient of proportionality. This method was first applied for vortices in superclean superconductors in Ref. [4] and since then was used widely (see for example, [5–7]). The resulting mass is of the order of the total mass of all particles within the area occupied by the vortex core. We will refer to this mass as to the dynamic mass. The dynamic mass originates from the inertia of excitations localized in the vortex core and can also be calculated as the momentum carried by localized excitations [8]. It is much larger and thus it is much more important than the mass obtained from the energy considerations. In the present Letter we develop a regular microscopic approach for calculating the dynamic vortex mass in a general case of a finite relaxation time τ of nonequilibrium excitations produced by the moving vortex. Using the Boltzmann kinetic equation for quasiparticles localized in the vortex core, we derive the equation for the vortex dynamics which contains the inertia term together with all the forces acting on a moving vortex. This approach is applied to both s- and d-wave superconductors. We find that dynamic mass displays a novel feature: it is a tensor whose components depend on the quasiparticle relaxation time. In s-wave superconductors, this tensor is diagonal in the superclean limit. The diagonal mass decreases rapidly as a function of the mean free time, and the off-diagonal components dominate in the moderately clean regime. In d-wave superconductors, the transition from a diagonal mass (in the extremely clean, high-τ limit) to an off-diagonal mass tensor (in the moderately clean, low-τ regime) occurs via the intermediate universal regime of the flux flow predicted in [9] (see also [10,11]), where both components are of the same order of magnitude. Our results agree with the previous work [4,5,8,11] in the limit τ → ∞. Forces on a moving vortex. The Boltzmann kinetic equation for quasiparticles localized in the vortex core is ∂f ∂t + ∂f ∂α ∂ǫn ∂� − ∂ǫn