Index Theorem and Vortex Kinetics in Bose-Einstein Condensates on a Haldane Sphere with a Magnetic Monopole

TL;DR

This paper explores the interplay of geometry and gauge fields in Bose-Einstein condensates on a Haldane sphere with a magnetic monopole, revealing an index theorem, vortex interactions, and scale-invariant dynamics.

Contribution

It introduces a novel index theorem linking vortices to gauge topology, and develops a kinetic theory for vortex dynamics in curved topological systems.

Findings

Established an index theorem connecting vortices and gauge topology

Derived universal logarithmic interactions between vortex-monopole composites

Predicted scale-invariant vortex dynamics confirmed by numerical simulations

Abstract

The geometry-gauge interplay constitutes a fundamental issue in quantum physics, with profound implications spanning from quantum gravity to topological matter. Here, we investigate the dynamic effects of geometry-gauge interplay in Bose-Einstein condensates (BECs) on a Haldane sphere with a magnetic monopole. We reveal an index theorem that establishes a correspondence between BEC vortices and the topology of the gauge field, enabling the construction of vortex-monopole composites. Furthermore, we derive the universal logarithmic interaction between composites, which governs the structure of the ground-state vortex lattice. By developing a kinetic theory, we predict scale-invariant vortex dynamics and an emergent duality. Both are confirmed through numerical simulations. This work first presents the dynamical coupling mechanism between spatial geometry and gauge fields, providing deep insights into superfluid systems with topological gauge structures in curved space.