Microscopic analysis of the nondissipative force on a line vortex in a superconductor: Berry’s phase, momentum flow, and the Magnus force

A microscopic analysis of the nondissipative force ${\mathbf{F}}_{\mathrm{nd}}$ acting on a line vortex in a type-II superconductor at T=0 is given. All work presented assumes a charged BCS superconductor. We first examine the Berry phase induced in the BCS superconducting ground state by movement of the vortex and show how this phase enters into the hydrodynamic action ${\mathit{S}}_{\mathrm{hyd}}$ of the superconducting condensate. Appropriate variation of ${\mathit{S}}_{\mathrm{hyd}}$ give ${\mathbf{F}}_{\mathrm{nd}}$ and variation of the Berry phase term is seen to contribute the Magnus or lift force of classical hydrodynamics to ${\mathbf{F}}_{\mathrm{nd}}$. This analysis, based on the BCS ground state of a charged superconductor, confirms in detail the arguments of Ao and Thouless within the context of the BCS model. Our Berry phase, in the limit e→0, is seen to reproduce the Berry phase determined by these authors for a neutral superfluid. We also provide a second, independent, determination of ${\mathbf{F}}_{\mathrm{nd}}$ through a microscopic derivation of the continuity equation for the condensate linear momentum. This equation yields the acceleration equation for the superflow and shows that the vortex acts as a sink for the condensate linear momentum. The rate at which momentum is lost to the vortex determines ${\mathbf{F}}_{\mathrm{nd}}$ in this second approach and the result obtained agrees identically with the previous Berry phase calculation. The Magnus force contribution to ${\mathbf{F}}_{\mathrm{nd}}$ is seen in both calculations to be a consequence of the vortex topology and motion. A preliminary discussion is given regarding finite temperature extensions of the Berry phase calculation, with emphasis on its relevance for the sign anomaly occurring in Hall-effect experiments on type-II superconductors in the flux flow regime.