Conserving and Gapless Approximations for an Inhomogeneous Bose Gas at Finite Temperatures

TL;DR

This paper derives and analyzes equations for a dilute Bose gas at finite temperatures, emphasizing conserving and gapless approximations, and discusses the spectrum of excitations within the Hartree-Fock-Bogoliubov framework.

Contribution

It provides a critical analysis of the HFB approximation, highlighting the gapless Popov variant and discussing the Beliaev second-order approximation for excitation spectra.

Findings

Popov approximation yields a gapless spectrum at all temperatures

The Beliaev second-order approximation relates to the HFB Green's function

Determining excitation spectra involves complex constraints

Abstract

We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self--consistent Hartree--Fock--Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg--Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single--particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.