Geometric phases of a vortex in a superfluid

We consider geometric phases of mobile quantum vortices in superfluid Bose-Einstein condensates. Haldane and Wu [Phys. Rev. Lett. 55, 2887 (1985)] showed that the geometric phase, $\gamma_{\mathcal C}=2\pi N_{\mathcal C}$, of such a vortex is determined by the number of condensate atoms $N_{\mathcal C}$ enclosed by the vortex trajectory. Considering an experimentally realistic freely orbiting vortex leads to an apparent disagreement with this prediction. We resolve it using the superfluid electrodynamics picture, which allows us to identify two additional contributions to the measured geometric phase; (i) a topologically protected edge current of vortices at the condensate boundary, and (ii) a superfluid displacement current. Our results generalise to, and pave the way for experimental measurements of vortex geometric phases using scalar and spinor Bose--Einstein condensates, and superfluid Fermi gases.