Microscopic Analysis of the Non-Dissipative Force on a Line Vortex in Superconductor

A microscopic analysis of the non-dissipative force F_nd acting on a line vortex in a type-II superconductor at T=0 is given. All work assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS ground state by movement of the vortex and show how this phase enters into the hydrodynamic action S_hyd of the condensate. Appropriate variation of S_hyd gives F_nd and variation of the Berry phase term is seen to contribute the Magnus force of classical hydrodynamics to F_nd. This analysis confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e -> 0, reproduces the Berry phase they obtain for a neutral superfluid. A second independent determination of F_nd is given through a microscopic derivation of the continuity equation for the condensate linear momentum. This yields the acceleration equation for the superflow. The vortex is seen to act as a momentum sink and the rate of momentum loss yields F_nd. Both calculations yield the same F_nd and show that the Magnus force contribution to F_nd is a consequence of the vortex motion and topology.