Finite-temperature excitations in a dilute Bose-condensed gas

Abstract

We present a systematic account of several approximations for the Beliaev self-energies for a uniform dilute Bose gas at finite temperature. We discuss the first-order Popov approximation, which gives a phonon velocity which vanishes as n 0(T) , where n 0( T) is the Bose condensate density. We generalize this Popov approximation using a t-matrix calculated with self-consistent ladder diagrams. This analysis shows that this t-matrix becomes very temperature-dependent and vanishes at T c, in agreement with Bijlsma and Stoof (1995). Finally, we make a detailed study of the Beliaev–Popov (B–P) approximation for the self-energies which are second order in the t-matrix. We work out the contribution from each individual diagram and give formal expressions for the self-energies and the excitation energy spectrum valid at arbitrary temperature. Careful attention is given to all infrared-divergent contributions. We rederive the well-known T=0 result of Beliaev (1958) for long-wavelength excitations. The analogous evaluation of the finite-temperature B–P self-energies is given in the low-frequency and long-wavelength limit. The corrections to the chemical potential, excitation energy and damping are all found to be proportional to the temperature near T c. All infrared divergent terms are shown to cancel out in physical quantities in the long-wavelength limit at arbitrary temperatures.