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Progress of Theoretical Physics

A Microscopic Theory of Superfluids. I: Quasi-Particles and the Extended Bogoliubov Transformation

Shuichiro Yamasaki

Published Sep 1, 1984• 3 citations
Publisher

Abstract

Creation and annihilation operators of quasi-particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non-zero momentum particles shows that the quasi-particles are phonons in the long wave-length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi-particles, approaches the Hartree-Fock state when the condensate is depleted completely at the critical temperature.

DOI
10.1143/ptp.72.441

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arXiv:cond-mat/0307745•Jul 30, 2003•1 citation
Creation and annihilation operators of quasi-particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non-zero momentum particles shows that the quasi-particles are phonons in the long wave-length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi-particles, approaches the Hartree-Fock state when the condensate is depleted completely at the critical temperature.
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.
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.
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.
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.