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Progress of Theoretical Physics

Use of a Coherent-State Representation for a Many-Boson System at High Density

Masahiko Tsujii, Toshiyuki Nishiyama

Published Jan 1, 1977• 0 citations
Publisher

Abstract

The ground-state energy of a many-boson system is obtained by the aid of a coherent-state representation in the high densigy limit. Dealing with the problem in a lattice space, some difficulties encountered in previous collective variable theories have been overcome. In particular, it is shown that Fröhlich's objection about the non-existence of the canonical conjugate to the density is not fatal to the previous theory of the density and phase operator approach. The ground-state energy obtained have agrees with that given by Brueckner and Lee.

DOI
10.1143/ptp.57.12

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Physical Review Letters•Apr 12, 1971•7 citations
The ground-state energy of a many-boson system is obtained by the aid of a coherent-state representation in the high densigy limit. Dealing with the problem in a lattice space, some difficulties encountered in previous collective variable theories have been overcome. In particular, it is shown that Fröhlich's objection about the non-existence of the canonical conjugate to the density is not fatal to the previous theory of the density and phase operator approach. The ground-state energy obtained have agrees with that given by Brueckner and Lee.
The momentum distribution function np is obtained for an interacting many-boson system at high density using the Rayleigh-Schrödinger perturbation theory in the scheme of the density and phase operator approach (DPO). The expression obtained has a correct form in the low momentum limit in accord with the condition first given by Gavoret and Nozières. The origin of an unphysical result found by Grest that np becomes negative for low wave numbers will be elucidated. Using the expression for np the ground-state energy is re-obtained. The result agrees completely with that given by Brueckner and Lee.
Green's functions of phonons for many-boson systems at high density are obtained from the phonon Hamiltonian given in a previous paper on the basis of the coherent-state representation. A dispersion equation derived from these Green's functions gives rise to the excitation spectrum leading to the Feynman energy in the low-momentum limit. It is shown that the dynamic structure function obtained here satisfies the sum rules in the low-momentum limit. As an illustrative example, the one-dimensional Lieb-Liniger model is considered. The relationships to the previous theories, in particular, to the velocity field approach of Yamasaki, Kebukawa and Sunakawa and to our density and phase operator approach are elucidated by the use of a unitary transformation. Discussion will be made on the D-wave portion of the roton-roton interaction in relation to the resonance state of roton pair studied in one of our previous papers. Finally, it is pointed out that the zeroth Fourier transform of the phase operator is related to the superfluid velocity in the two-fluid theory of liquid helium II.
The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.
Concrete expressions are given to the operators φk forming a set of canonically conjugate variables to the density fluctuations ρl in a Bose-Einstein condensed many-boson system. The Fourier transform of φk to the coordinate space admits in classical terms an interpretation as the phase of the boson field ψ(x). Various objections against the existence of canonically conjugate variables to the density are critically examined. They do not apply to the proposed operator φk which is non local and canonically conjugate only to the density fluctuation.