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Physical Review A

Ground State of an Interacting Bose Gas

Paul Berdahl, Deok Kyo Lee

Published Apr 1, 1973• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.7.1376

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Progress of Theoretical Physics•Aug 1, 1976•2 citations
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.
Several expression which are apparently different from each other have been derived for the first correction to Bogoliubov's expression for the ground-state energy of a Bose system. The relationship between these expressions is investigated. It is found that the expressions of Hiroike, Brueckner-Lee and Lee are equivalent to each other. It is also found that the expressions of Bogoliubov-Zubarev and Takahashi, which are equivalent to each other, do not always agree with the above-mentioned expressions. A sufficient condition for the interaction potential is derived in order that such a disagreement does not arise.
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 validity of Sunakawa, Yamasaki and Kebukawa's Hamiltonian and that of Bogoliubov and Zubarev's Hamiltonian are examined. Perturbational expansion of the ground state energy by these Hamiltonians disagrees with the exact solution of Lieb and Liniger for one-dimensional Bose system with repulsive delta-function interaction. This fact suggests that these Hamiltonians are not microscopic descriptions of the many-Boson system. Mathematical inconsistency in Bogoliubov and Zubarev's theory is also pointed out. Moreover analytic expression of high density expansion for the ground state energy density e0 is found out to be e0n-3=γ-(4/3π)γ3/2+(1/6-1/π2)γ2+O(γ5/2), γ≡c/n, for one-dimensional Bose system with delta function interaction (density n, strength 2c, ħ=2m=1) by the use of the correlated basis function method.
Ground states of two typical many-boson model systems are studied: (i) a weakly interacting boson system at low density, and (ii) a low-density hard-sphere boson gas in the low-energy limit. For both of these systems it is shown explicitly that the uniform-limit formalism based on the method of correlated basis functions can be used to obtain, in a relatively simple way, exact results for the ground-state energy per particle and other related properties in the weak-coupling limit. In the problem of the hard-sphere boson gas the TTT-matrix method is found to be particularly useful in dealing with the singular interaction potential.
A new theory of the interacting bose system is proposed. In contrast with the Bogoliubov theory, the condensate in the present theory is composed of the interacting dressed atoms and the total number of the dressed atoms is conserved. The second-order corrections to the ground-state energy and the excitation energy are evaluated to be found that the results in the present theory coincide with those obtained by Sunakawa et al. On the basis of this situation, it is shown that the present formalism is essentially equivalent to the theory of Sunakawa et al.