Microscopic Analysis of the Non-Dissipative Force on a Line Vortex in a Superconductor: Berry's Phase, Momentum Flows and the Magnus Force

A microscopic analysis of the non-dissipative force ${\bf F}_{nd}$ acting on a line vortex in a type-II superconductor at $T=0$ is given. We first examine the Berry phase induced in the true superconducting ground state by movement of the vortex and show how this induces a Wess-Zumino term in the hydrodynamic action $S_{hyd}$ of the superconducting condensate. Appropriate variation of $S_{hyd}$ gives ${\bf F}_{nd}$ and variation of the Wess-Zumino term is seen to contribute the Magnus (lift) force of classical hydrodynamics to ${\bf F}_ {nd}$. This first calculation confirms and strengthens earlier work by Ao and Thouless which was based on an ansatz for the many-body ground state. We also determine ${\bf F}_{nd}$ 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 ${\bf F}_{nd}$ and the result obtained agrees with the Berry phase calculation. The Magnus force contribution to ${\bf F}_{nd}$ is seen to be a consequence of the vortex topology. Preliminary remarks are made regarding finite temperature extensions, with emphasis on its relevance to the sign anomaly occurring in Hall effect experiments done in the flux flow regime.