# Bose-Einstein condensation in two-dimensional traps

## TL;DR

This paper develops an analytical method to accurately determine the critical temperature and condensate fraction for Bose-Einstein condensation in two-dimensional traps, addressing divergence issues and validating results with numerical calculations.

## Contribution

It introduces an analytical continuation approach to resolve divergence problems in 2D BEC studies and provides improved expressions for critical parameters.

## Key findings

- Analytical expressions for critical temperature and condensate fraction in 2D traps.
- Good agreement between analytical results and numerical calculations.
- Validation of grand canonical ensemble calculations for 2D BEC.

## Abstract

In two-dimensional traps, since the theoretical study of Bose-Einstein condensation (BEC) will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a two-dimensional anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.