Effective-action approach to a trapped Bose gas

TL;DR

This paper applies the effective-action formalism to analyze a trapped Bose gas, deriving equations for condensate and excitations, and clarifies the validity of Bogoliubov and Popov approximations in this context.

Contribution

It introduces an effective-action approach to describe trapped Bose gases, connecting loop expansion with known approximations and analyzing their validity.

Findings

One-loop order reproduces Bogoliubov equations for trapped gases.

Validity of Bogoliubov approximation depends on trap size exceeding scattering length.

Analytical and numerical analysis of the effective-action formalism for dilute Bose gases.

Abstract

The effective-action formalism is applied to a gas of bosons. The equations describing the condensate and the excitations are obtained using the loop expansion for the effective action. For a homogeneous gas the Beliaev expansion in terms of the diluteness parameter is identified in terms of the loop expansion. The loop expansion and the limits of validity of the well-known Bogoliubov and Popov equations are examined analytically for a homogeneous dilute Bose gas and numerically for a gas trapped in a harmonic-oscillator potential. The expansion to one-loop order, and hence the Bogoliubov equation, is shown to be valid for the zero-temperature trapped gas as long as the characteristic length of the trapping potential exceeds the s-wave scattering length.