Spectral Flow, the Magnus Force, and the Josephson-Anderson Relation

TL;DR

This paper demonstrates that spectral flow in vortices is equivalent to phase slippage, confirming the Berry phase calculation of the Magnus force and clarifying misconceptions about force cancellation.

Contribution

It establishes the equivalence between spectral flow and phase slippage, and clarifies the distinct roles of spectral flow and Magnus force in superfluid vortices.

Findings

Spectral flow equals phase slippage process.

Spectral flow confirms Berry's phase Magnus force.

Localized core states do not contribute to spectral flow.

Abstract

Based on a recent work (Thouless, Ao, and Niu, Transverse force on a quantized vortex in a superfluid) I show that the spectral or momentum flow due to a moving vortex is identical to the phase slippage process discussed by Josephson and Anderson. I also show that the force due to the spectral flow is the reaction force of the Magnus force, and that the two forces act on different objects. Therefore one can conclude that the evaluation of the spepctral flow confirms the results by the Berry's phase calculation of the Magnus force, not the cancelation of the Magnus force as claimed by some authors. A by-product of the present proof is that localized core states do not contribute to the spectral flow, as also noted in the work of Thouless, Ao, and Niu on the Magnus force.