The saddle-point method for condensed Bose gases

TL;DR

This paper introduces a modified saddle-point method for accurately analyzing condensed Bose gases across all temperatures, including the critical point, overcoming limitations of the conventional approach.

Contribution

A novel saddle-point approximation that properly handles the ground-state singularity, enabling precise statistical analysis of Bose gases in all regimes.

Findings

The new method accurately captures the Bose-Einstein condensation onset.

It reveals a universal error in the conventional approximation within the condensate regime.

The approach provides analytical insights into the transition from thermal to condensed phases.

Abstract

The application of the conventional saddle-point approximation to condensed Bose gases is thwarted by the approach of the saddle-point to the ground-state singularity of the grand canonical partition function. We develop and test a variant of the saddle-point method which takes proper care of this complication, and provides accurate, flexible, and computationally efficient access to both canonical and microcanonical statistics. Remarkably, the error committed when naively employing the conventional approximation in the condensate regime turns out to be universal, that is, independent of the system's single-particle spectrum. The new scheme is able to cover all temperatures, including the critical temperature interval that marks the onset of Bose--Einstein condensation, and reveals in analytical detail how this onset leads to sharp features in gases with a fixed number of particles. In particular, within the canonical ensemble the crossover from the high-temperature asymptotics to the condensate regime occurs in an error-function-like manner; this error function reduces to a step function when the particle number becomes large. Our saddle-point formulas for occupation numbers and their fluctuations, verified by numerical calculations, clearly bring out the special role played by the ground state.