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Progress of Theoretical Physics

Excitation-Modes and Their Interactions in a Many-Boson System. I

Shuichiro Yamasaki, T. Kebukawa, Sigenobu Sunakawa

Published Apr 1, 1979• 2 citations
Publisher

Abstract

In previous papers, it has been shown that the new excitation-mode operators bk and bk* are appropriate to describe the phonon-phonon interactions in a boson system. In this paper, the canonical transformation from the old operators Bκ and Bκ* to the new ones is determined up to order of N-3/2 by making use of the “asymptotic one-momentum-transfer structure” of the density-fluctuation operator and the momentum-density operator.

DOI
10.1143/ptp.61.996

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Physical Review A•Mar 1, 1972•3 citations
In previous papers, it has been shown that the new excitation-mode operators bk and bk* are appropriate to describe the phonon-phonon interactions in a boson system. In this paper, the canonical transformation from the old operators Bκ and Bκ* to the new ones is determined up to order of N-3/2 by making use of the “asymptotic one-momentum-transfer structure” of the density-fluctuation operator and the momentum-density operator.
Abstract The comment is based on misunderstandings and, as a result, does not evaluate the norm operator method. We refute the comment. The structure of the norm operator composed of all the excitation modes determines whether the form of a practical boson expansion is a small parameter expansion. KT-3 is chimerical. It should be taken seriously that the small parameter expansions do not hold with truncating the contributions of all phonon excitation modes that are not adopted as boson excitations. The formulation using the boson expansion of the projection operator on to the boson vacuum does not allow the limitation of the phonon excitation number beforehand.
A new well-defined velocity operator is proposed for the collective description of a many-boson system. It is shown that the new velocity operator is irrotational. The Hamiltonian of the system is successfully described in terms of the density-fluctuation operator and the velocity operator. The interaction term in the order of N-1 has a different form from that in previous papers. The range of validity of the present formalism is discussed in the final section in answer to the questions raised so far on the collective formalism of the many-boson system.
The convolution approximations to the three• particle and four-particle structure functions are incorporated in a non-linear canonical transformation applicable to a system of degenerate bosons.The transformation relates the quasi• particle operators bk to the density fluctuations P k and their canonical conjugates 'P k by way of a formal series in powers of N-ll ', where N is the number of particles.Higher order terms in the series require corrections to the convolution approximations.The next order terms are calculated to improve the approximations to the structure functions.The present formalism provides an unambiguous way of evaluating the residual interaction not taken account of in the trial state, the b•vacuum, which admits the convolution approximation or its extension. § 1. Introduction
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.
As a continuation of a recent paper, using density and current density as coordinates for interacting bosons at absolute zero, this work deals with the perturbation theory. The zerothorder Hamiltonian is the one shown to yield the Bogoliubov energy spectrum. First- or secondorder perturbation calculations have been made of the three-phonon and the four-phonon vertices for the energy corrections of the low-lying excited states. The results in the long-wavelength limit differ, although only slightly, from the quantum hydrodynamic calculations, and indicate that the perturbed excitation energy ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is an odd-power series in kkk if the Fourier coefficients of the two-body potential concerned can be approximated by a constant or an evenpower series in kkk. However, whether this remains so in all higher orders of various perturbations is an open question. Two methods were used in the calculation. The first involves functional representations of the coordinates in the ${\ensuremath{\rho}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ representation recently reported. The second method is new, and employs "occupation-number" representations of the current algebra by introducing creation and annihilation operators for "phonons" from density fluctuations. These latter representations enable us to benefit from the advantages of the canonical field theory.