Vortices in Superfluid Films on Curved Surfaces

TL;DR

This paper investigates how the Gaussian curvature of curved surfaces influences vortex behavior in superfluid films, providing both perturbative and exact solutions for the geometric potential affecting vortex interactions.

Contribution

It introduces a non-perturbative conformal mapping method to exactly compute the geometric potential and vortex interactions on curved surfaces, extending previous perturbative approaches.

Findings

Gaussian curvature acts as a source for vortex attraction or repulsion.

Exact solutions for geometric potential are obtained via conformal mappings.

Universal bounds on the strength of geometric effects are derived.

Abstract

We present a systematic study of how vortices in superfluid films interact with the spatially varying Gaussian curvature of the underlying substrate. The Gaussian curvature acts as a source for a geometric potential that attracts (repels) vortices towards regions of negative (positive) Gaussian curvature independently of the sign of their topological charge. Various experimental tests involving rotating superfluid films and vortex pinning are first discussed for films coating gently curved substrates that can be treated in perturbation theory from flatness. An estimate of the experimental regimes of interest is obtained by comparing the strength of the geometrical forces to the vortex pinning induced by the varying thickness of the film which is in turn caused by capillary effects and gravity. We then present a non-perturbative technique based on conformal mappings that leads an exact solution for the geometric potential as well as the geometric correction to the interaction between vortices. The conformal mapping approach is illustrated by means of explicit calculations of the geometric effects encountered in the study of some strongly curved surfaces and by deriving universal bounds on their strength.