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Low Temperature Physics

Another approach for obtaining the excitation spectra in degenerate Bose gases with delta-shaped interaction potentials

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

Published Mar 1, 2017• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.4979959

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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.
Within the self-consistent Hartree–Fock approximation, the equilibrium weakly nonideal Bose gas with a delta-shaped interaction potential in the presence of the Bose–Einstein condensate is considered without using quasi-averages. On this basis, using the virial theorem and diagram techniques of the perturbation theory for the equilibrium system in a macroscopic volume, the equation of state providing the isothermal compressibility finiteness, including the Bose–Einstein condensate domain of existence, is obtained.
An NNN-body Bose-Einstein system of particles with long-range attraction and hard-sphere repulsion between particles is considered. It is shown that if the constants of the interaction have values within a certain range it is possible to calculate the ground-state energy of the system as a function of $\frac{\ensuremath{\Omega}}{N}$, where $\ensuremath{\Omega}$ is the volume of the box containing the system, in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}\ensuremath{\rightarrow}\ensuremath{\infty}$, with $\frac{N}{\ensuremath{\Omega}}=\ensuremath{\rho}$ fixed. The results show that the system can possess an NNN-body bound state, which has an equilibrium density and negative energy, and that the interactions can be saturating. Excited states are also considered. It is shown that low-lying excitations consist purely of phonons, whose velocity agrees with that computed from the macroscopic compressibility, furnished by the ground-state energy. The formula for the general excited energy levels suggests that thermodynamically the system may have a "gas" phase and two "liquid" phases, the transition between the two "liquid" phases being the analog of the Bose-Einstein condensation of the ideal gas. Thermodynamic considerations are, however, not contained in this paper.
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.
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.
We present a method of finding approximate analytical solutions for the spectra and eigenvectors of collective modes in a two-dimensional system of interacting bosons subjected to a linear external potential or the potential of a special form $u(x,y)=\ensuremath{\mu}\ensuremath{-}u{\mathrm{cosh}}^{2}x/l,$ where \ensuremath{\mu} is the chemical potential. The eigenvalue problem is solved analytically for an artificial model allowing the unbounded density of the particles. The spectra of collective modes are calculated numerically for the stripe, the rare density valley, and the edge geometry and compared with the analytical results. It is shown that the energies of the modes localized at the rare density region and at the edge are well approximated by the analytical expressions. We discuss Bose-Einstein condensation (BEC) in the systems under investigations at $T\ensuremath{\ne}0$ and find that in case of a finite number of the particles the regime of BEC can be realized, whereas the condensate disappears in the thermodynamic limit.