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Physical Review A

Microscopic calculation of the excitation spectrum of superfluid helium-4

Gerald W. Goble, Donald H. Kobe

Published Sep 1, 1974• 11 citations
Publisher

Abstract

The Hartree-Fock-Bogoliubov equations for a homogeneous boson system at finite temperature are derived using linearized operator equations of motion. By a partial decoupling of two of the equations, a temperature-dependent form of the Lippmann-Schwinger equation for a local $t$ matrix emerges naturally with a fully dressed energy denominator. In order to use a realistic potential the theory is phenomenologically modified by (i) redefining the chemical potential to remove an unphysical energy gap, and (ii) removing (possibly divergent) Hartree-Fock-like terms from the single-particle energy. The resulting set of coupled nonlinear integral equations is solved self-consistently for the Morse-dipole-dipole and the Frost-Musulin potentials. The excitation spectra obtained are in qualitative agreement with the observed phonon-roton spectrum, but the temperature dependence is given incorrectly. The calculations were made at a realistic density and the condensate fraction is about 60% at absolute zero.

DOI
10.1103/physreva.10.851

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Physical Review A•Jun 1, 1973•36 citations
The Hartree-Fock-Bogoliubov equations for a homogeneous boson system at finite temperature are derived using linearized operator equations of motion. By a partial decoupling of two of the equations, a temperature-dependent form of the Lippmann-Schwinger equation for a local ttt matrix emerges naturally with a fully dressed energy denominator. In order to use a realistic potential the theory is phenomenologically modified by (i) redefining the chemical potential to remove an unphysical energy gap, and (ii) removing (possibly divergent) Hartree-Fock-like terms from the single-particle energy. The resulting set of coupled nonlinear integral equations is solved self-consistently for the Morse-dipole-dipole and the Frost-Musulin potentials. The excitation spectra obtained are in qualitative agreement with the observed phonon-roton spectrum, but the temperature dependence is given incorrectly. The calculations were made at a realistic density and the condensate fraction is about 60% at absolute zero.
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.
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.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
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 investigation is performed using normal-mode analysis on a quantum mechanical cell model whose general features are representative of 4He at low temperatures. Both random-phase and self-consistent field approximations are invoked. A Bogoliubov-type excitation energy spectrum is derived for the interacting Bose system, and the amplitudes of the system quasi-particles are obtained; these quantities are expressed in terms of zero-momentum occupation number. A relationship giving the temperature-dependent fractional occupation of the single-particle zero-momentum level is established, and special attention is devoted to the temperature extreme near absolute zero. An estimate is made of the zero-temperature sound speed corresponding to the special case of phonon excitation given by the model in low momentum. It is shown that at a critical temperature for which the zero-momentum occupation is depleted, the quasi-particles indeed become normal Bose particles. In first approximation, this critical temperature is akin in form to that obtained in treatments of ideal boson systems.
By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.