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arXiv:nucl-th/9907088

Hartree--Fock--Bogoliubov Approximation for Finite Systems

Aurel Bulgac

Published Jul 22, 1999• 18 citations
arXivHTMLPDF

Abstract

Some general features of the spectrum of the Hartree-Fock-Bogoliubov equations are examined. Special attention is paid to the asymptotic behavior of the single quasiparticle wave functions (s.qp.w.fs.), matter density distribution and density of the pair condensate. It is shown that due to the coupling between hole and particle states,the deeply bound hole states acquire a width and have to be treated as continuum states. The proper normalization of the s.qp.w.fs. is discussed.

DOI
10.48550/arxiv.nucl-th/9907088

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Some general features of the spectrum of the Hartree-Fock-Bogoliubov equations are examined. Special attention is paid to the asymptotic behavior of the single quasiparticle wave functions (s.qp.w.fs.), matter density distribution and density of the pair condensate. It is shown that due to the coupling between hole and particle states,the deeply bound hole states acquire a width and have to be treated as continuum states. The proper normalization of the s.qp.w.fs. is discussed.
We present a generalization of the Hartree-Fock Bogoliubov (HFB) theory in which the coupling between one and two quasi-particles is taken into account.This is done by writing the excitation operators as linear combinations of one and two HFB quasi-particles. The excitation energies and the quasi-particle amplitudes are given by generalized Bogoliubov equations. The excitation spectrum has two branches. The first one is a discrete branch which is gapless and has a phonon character at large wave-length and, contrarily to HFB, is always stable. This branch is detached from a continuum branch whose threshold, at fixed total momentum, coincides with the two quasi-particle threshold of the HFB theory. The gap between these two branches at P=0 is equal to two times the HFB gap, which then provides for the relevant energy scale. We also give numerical results for a specific case.
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.
An extension of the Hartree–Fock–Bogoliubov (HFB) theory of degenerate Bose systems in which the coupling between one and two quasi-particles is taken into account is developed. The excitation operators are written as linear combinations of one and two HFB quasi-particles. Excitation energies and quasi-particle amplitudes are given by generalized Bogoliubov equations. The excitation spectrum has two branches. The first one is a discrete branch which is gapless and has a phonon character at large wavelength and, contrarily to HFB, is always stable. This branch is detached from a second, continuum branch whose threshold, at fixed total momentum, coincides with the two quasi-particle threshold of the HFB theory. The gap between the two branches at P = 0 is twice the HFB gap, which thus provides for the relevant energy scale. Numerical results for a specific case are given.
We critically examine the Hartree-Fock-Bogoliubov (HFB) solution of the equations of motion for condensate fluctuations in a weakly coupled plasma of charged bosons at finite temperature. Analytic expressions are derived for the first two infrared-divergent terms in both the momentum distribution of the noncondensate and the anomalous Bose correlation function at low momenta. Incorporation into the theory of the appropriate form of the Hugenholtz-Pines relation for the chemical potential is needed to cancel an unphysical divergence. Exact cancellation of infrared-divergent terms is demonstrated in the HFB shift of the single-particle excitation energy away from the Bogoliubov value at long wavelengths, with the residual terms raising it towards the plasma frequency at low temperature. Numerical illustrations are presented for a number of properties of the boson plasma as functions of temperature and density in the weak-coupling regime: these are the chemical potential, the condensate fraction, the normal and anomalous momentum distribution functions and the corresponding one-body density matrices, and the dispersion relation of single-particle excitations.
Pair wave functions (PWF) which are eigenfunctions of the reduced density 2-matrix are considered for a homogeneous Bose liquid. With the Bogoliubov principle of the correlation weakening it is proved that the distribution of the "dissociated" pair states over momenta is exactly the product of the single-particle distribution functions. Thus, the "dissociated" pair states are naturally classified as condensate-condensate, condensate-supracondensate and supracondensate-supracondensate ones provided the Bose-Einstein condensate exists. The condensate-condensate as well as condensate-supracondensate PWF are expressed in terms of the averages of products of the creation and destruction Bose operators. This leads to the simple interpretation of the anomalous averages as the "scattering parts" of the condensate-condensate and condensate-supracondensate PWF. It is shown that in contrast to the Fermi liquid, the appearance of the anomalous averages for the Bose liquid does not necessarily mean that there exist bound states of pairs of particles. The PWF in the Hartree-Fock-Bogoliubov (HFB) approach are found. Given the density of the condensate is not zero, there are no bound pair states in the HFB scheme. The expansion of the pair correlation function in the set of PWF is very useful in order to take into account both short-range and long-range spatial correlations. Applications (possible and already realized) of the formalism developed are discussed.
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.