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Frontiers of Physics

On the ground state energy of the inhomogeneous Bose gas

Viktor Borisovich Bobrov, S. A. Trigger

Published Mar 6, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.1007/s11467-017-0624-7

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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.
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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.
We present a self-consistent ttt-matrix theory for Bose-Einstein-condensed systems within the Hartree-Fock-Bogoliubov (HFB) approximation. Using the Lippmann-Schwinger equation for a ttt matrix describing the collision between two particles via an interparticle potential, we derive a set of equations for the normal and anomalous self-energies in the HFB approximation expressed in terms of the ttt matrix. These equations are solved for a hard-sphere potential. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low densities and to the Brueckner-Sawada theory at high densities. The spectrum is gapless and linear in small momentum, but does not have any roton minimum in the large-momentum region even for high densities such as those of 4{}^{4}4He.