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Physical Review A

Energy Spectrum of Elementary Excitations in a Boson Fluid

Deok Kyo Lee

Published Nov 1, 1973• 6 citations
Publisher

Abstract

The Bogoliubov-Zubarev method of collective coordinates is used to derive the energy spectrum of elementary excitations in a weakly interacting many-boson system through second-lowest order in a weak coupling expansion. The obtained result of the spectrum is shown to be equivalent to that evaluated by means of the variation-perturbation procedure based on the method of correlated basis functions in the uniform limit. Comparison of the excitation spectrum is also made briefly with two other results calculated using the Bogoliubov-Zubarev Hamiltonian in somewhat related but different approaches.

DOI
10.1103/physreva.8.2735

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The Bogoliubov-Zubarev method of collective coordinates is used to derive the energy spectrum of elementary excitations in a weakly interacting many-boson system through second-lowest order in a weak coupling expansion. The obtained result of the spectrum is shown to be equivalent to that evaluated by means of the variation-perturbation procedure based on the method of correlated basis functions in the uniform limit. Comparison of the excitation spectrum is also made briefly with two other results calculated using the Bogoliubov-Zubarev Hamiltonian in somewhat related but different approaches.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.
The ground state of a many-particle boson system is studied for two closely related limits: the uniform limit and the weak coupling limit. The former is defined by $\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1$ and the latter by $\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1$, where g(r)g(r)g(r) is the radial distribution function and N0{N}_{0}N0​ is the mean occupation number of the zero-momentum state. In the uniform limit the variation-perturbation approach based on (a) the method of correlated basis functions and (b) the series expansion in powers of $\ensuremath{\alpha}$ is found to be equivalent to the field-theoretic treatment given by Brueckner (for the charged-boson gas) in the weak coupling limit. In particular, it is shown that the variation-perturbation energy obtained for the uniform limit in the momentum representation is identical through second order to the ground-state energy evaluated for $\ensuremath{\beta}\ensuremath{\ll}1$ by summing one- and two-ring diagrams in the Bogoliubov occupation-number representation. The charged-boson gas and the one-dimensional boson system with a $\ensuremath{\delta}$-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
Presently there are two distinct classes of microscopic theories which describe a system of many identical spinless bosons. One is the Bogoliubov theory, which places special emphasis on the occupation of the zero-momentum state and as such depends explicitly on the number of particles in this state, N0{N}_{0}N0​. The other depends on density fluctuations to describe the fluid. In this work, we derive a modified version of the Bogoliubov theory which results in a number-conserving Hamiltonian with no explicit dependence on N0{N}_{0}N0​. The ground- and first-excited-state energies evaluated through second order in 1N12\frac{1}{{N}^{\frac{1}{2}}}N21​1​ are equivalent to the results of a variational-perturbative procedure based on the method of correlated basis functions in the uniform limit.
The method of “auxiliary variables” (collective coordinates) used by Bogolyubov and Zubarev is formulated in a somewhat different manner. Improvements are made in two points: 1) The Hamiltonian is given in an Hermitian form and 2) the number of the auxiliary variables can be limited, if necessary. We calculate the ground state energy and the excitation energy spectrum of the spinless bosons taking account of phonon-phonon interactions. The expression for the ground state energy is shown to be equivalent to the one derived by Bogolyubov and Zubarev.
An occupation-number formulation is considered for a many-particle boson system when the ground state is described by the Bijl-Dingle-Jastrow (BDJ) type of variational wave function. A nonunitary operator acting on the state function of a noninteracting boson system is introduced to convert the BDJ function in configuration space into an equivalent state function in second-quantized space. The momentum distribution function is derived in an infinite series in terms of the liquid-structure function and the correlation function. In the case of the optimum variational description under the uniform limit, the leading term of the obtained result is found to agree with the Bogoliubov formula evaluated in the weak-coupling limit. The same analysis in the occupation-number space also reproduces the mean kinetic energy calculated in configuration space.