A gapless theory of Bose-Einstein condensation in dilutegases at finite temperature

Abstract

In this paper we develop a gapless theory of Bose-Einsteincondensation (BEC) which can beapplied to both trapped and homogeneous gases at zero and finitetemperature. The starting point for the theory is the secondquantized, many-body Hamiltonian for a system of structurelessbosons with pairwise interactions. A number-conserving approachis used to rewrite this Hamiltonian in a form which isapproximately quadratic with higher-order cubic and quarticterms. The quadratic part of the Hamiltonian can be diagonalizedexactly by transforming to a quasiparticle basis, whilerequiring that the condensate satisfy the Gross-Pitaevskiiequation. The non-quadratic terms are assumed to have a smalleffect and are dealt with using first- and second-orderperturbation theory. The conventional treatment of these terms,based on factorization approximations, is shown to beinconsistent.Infrared divergences can appear in individual terms of theperturbation expansion, but we show analytically that the totalcontribution beyond quadratic order is finite. The resultingexcitation spectrum is gapless and the energy shifts are smallfor a dilute gas away from the critical region, justifying theuse of perturbation theory. Ultraviolet divergences can appearif a contact potential is used to describe particleinteractions. We show that the use of this potential as anapproximation to the two-body T-matrix leads naturally to ahigh-energy renormalization.The theory developed in this paper is therefore well defined atboth low and high energy and provides a systematic descriptionof BEC in dilute gases. It can thereforebe used to calculate the energies and decay rates of theexcitations of the system at temperatures approaching the phasetransition.