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Canadian Journal of Physics

NORMAL-MODE ANALYSIS OF A SUPERFLUID MODEL

G. C. Knollman

Published Sep 1, 1967• 1 citation
Publisher

Abstract

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.

DOI
10.1139/p67-261

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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.
The low-temperature properties of the interacting Bose system at high density are investigated under the random phase approximation and the self-consistent Hartree approximation based on the “normal-mode analysis”. To treat rigorously the tempereture-dependent “depletion” effect of the particles in the zero-momentum level, a new approach is proposed. As a result, the critical temperature, Tc, at which the zero-momentum particles disappear is determined. In the region below Tc, the temperature-dependent energy-spectrum of the quasi-particles, which has at the zero-temperature an intimate connection with that of Bogoliubov, is obtained, and the property of superfluidity which disappears at Tc is discussed. It is shown that the quasi-particles are transformed into the normal Bose particles at Tc, and the temperature-dependent energy-spectrum of the normal particles in the region above Tc is determined. It is noteworthy that the phase transition of the second-order occurs at Tc. In so far as the interaction gives the real and positive value for the energy-spectrum of both the quasi-particles and normal particles, the results above hold true, independent of the magnitude and particular shape of interaction.Some remarks are made on the hard-sphere interaction and on the interaction among the quasi-particles which seems necessary for the investigateions of the intermediate-density Bose system.
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Study of the temperature dependence of single-particle and pair coherent condensate densities for a Bose liquid with a "depleted" single-particle Bose-Einstein condensate (BEC) at T≠0T \neq 0T=0
We study the temperature dependence of single-particle and pair coherent condensate densities for a Bose liquid with a "depleted" single-particle Bose-Einstein condensate (BEC) at T≠0T \neq 0T=0. Our investigations were based on the field-theoretical Green's functions. On the basis of the use of empirical data about the speeds of the first and second sounds is described superfluid state at T≠0T \neq 0T=0. It is studied the structure of superfluid state taking into account appearance with the different from zero temperatures of normal component and taking into account the branch of the second sound, whose speed approaches zero at T→TλT \to T_λT→Tλ​. The bare interaction between bosons was chosen in the form of a repulsive Azis potential, the Fourier component of which was an oscillatory and sign-varying function of the momentum. The obtained temperature dependence of single-particle and pair coherent condensate and total densities has a good agreement with experiment data.
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.
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.