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Low Temperature Physics

On the ground-state energy of a finite inhomogeneous degenerate Bose gas

Viktor Borisovich Bobrov, Anatolij G. Zagorodny, S. A. Trigger

Published Nov 1, 2018• 2 citations
Publisher

Abstract

The ground-state energy of a finite inhomogeneous system of bosons located in a scalar external field was found within the framework of the self-consistent Hartree-Fock approximation, on the basis of the representation of the second quantization without using the formalism of anomalous averages. The ground-state wave function corresponds to the stationary Gross-Pitaevskii equation for the Bose-Einstein condensate wave function. It was shown that the ground-state energy can be found using the energy determined by the stationary Gross-Pitaevskii equation only for a system that satisfies the thermodynamic limit.

DOI
10.1063/1.5062162

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The ground-state energy of a finite inhomogeneous system of bosons located in a scalar external field was found within the framework of the self-consistent Hartree-Fock approximation, on the basis of the representation of the second quantization without using the formalism of anomalous averages. The ground-state wave function corresponds to the stationary Gross-Pitaevskii equation for the Bose-Einstein condensate wave function. It was shown that the ground-state energy can be found using the energy determined by the stationary Gross-Pitaevskii equation only for a system that satisfies the thermodynamic limit.
Within the self-consistent Hartree–Fock approximation, an explicit in this approximation expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose–Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross–Pitaevski approach, is found. The obtained form of kinetic energy is valid beyond the Hartree–Fock approximation and can be applied for arbitrary strong interparticle interaction.
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.
A formalism is proposed for describing an equilibrium Bose gas based on consideration of a macroscopic but finite volume. The presence of a Bose-Einstein condensate is taken into account through an off-diagonal long-range order in the transition to the thermodynamic limit. On this basis, degenerate Bose gases with delta-shaped interaction potentials are described in terms of a self-consistent Hartree-Fock approximation. Explicit expressions are obtained for the energy spectra of single-particle and collective excitations.
Based on the Schrodinger equation, exact expressions for the non-relativistic particle energy in the local external field and the external field potential are derived as inhomogeneous density functionals. On this basis, it is shown that, when considering more than two noninteracting electrons, the energy of such a system cannot be an inhomogeneous density functional. The result is extended for the system of interacting electrons. This means that the Hohenberg-Kohn lemma which assert that in the ground state to each inhomogeneous density corresponds only one potential of the external field cannot be a justification of the existence of the universal density functional in the general case. At the same time, statements of the density functional theory remain valid when considering any number of noninteracting ground-state bosons due to the Bose condensation effect.