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Progress of Theoretical Physics

Ground-State Energy of a Bose System in the Weak Coupling Limit

Kazuo Hiroike

Published Aug 1, 1975• 5 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.54.308

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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 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 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.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
The calculation of the ground-state energy of a homogeneous Bose gas with soft interparticle interaction results in an expression originally stated by Brueckner and Sawada, instead of the more accurate Lee-Yang formula. We show that the source of this discrepancy lies in the residual terms of the full Hamiltonian which are neglected in the standard Bogoliubov scheme. Incorporating these residual interactions, we derive an improved upper bound on the ground-state energy. An important link in our chain of arguments consists in the fact that the Born series of the scattering length is a convergent rather than an asymptotic expansion at least for su ciently soft potentials. We provide an exactly solvable model which demonstrates that the Born series of the scattering length reproduces, within its radius of convergence, the exact scattering length.
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.
The textbook calculation of the ground-state energy of a dilute gas of Bose particles is examined in detail, and certain mathematical inconsistencies are pointed out. On the basis of this analysis, a refined approach suitable for soft interaction potentials which lend themselves to a low-order Born approximation is developed. This procedure emphasizes the low-density character of the resulting formula for the ground-state energy, and avoids all divergent expressions at intermediate stages of the computation. It is stressed that the standard Bogoliubov approximation, if not augmented by some additional device, leads to an error which manifests itself already in the lowest order of the density. - PACS 03.75.Hh, 05.30.Jp, 03.65.Nk.