# Bose-Einstein Condensation on the Surface of a Sphere

## TL;DR

This paper explores Bose-Einstein condensation on a spherical surface, deriving critical temperature and condensate fraction analytically, and examines the effects of finite size and interactions, including the BKT transition.

## Contribution

It provides the first analytical study of BEC thermodynamics on a spherical surface, including effects of interactions and finite size, and analyzes the BKT transition in this geometry.

## Key findings

- Critical temperature and condensate fraction derived analytically.
- Finite sphere radius influences BEC behavior, recovering 2D results at infinite radius.
- BKT transition analyzed on the spherical surface.

## Abstract

Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.