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Physical Review A

Ground-State Energy of a Many-Particle Boson System

Deok Kyo Lee

Published Oct 1, 1971• 29 citations
Publisher

Abstract

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 $g(r)$ is the radial distribution function and ${N}_{0}$ 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.

DOI
10.1103/physreva.4.1670

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Physical Review•Feb 1, 1965•79 citations
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 \ensuremathα=1\ensuremath−g(0)\ensuremath≪1\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremath{\ll}1\ensuremathα=1\ensuremath−g(0)\ensuremath≪1 and the latter by \ensuremathβ=(N\ensuremath−N0)N\ensuremath≪1\ensuremath{\beta}=\frac{(N\ensuremath{-}{N}_{0})}{N}\ensuremath{\ll}1\ensuremathβ=N(N\ensuremath−N0​)​\ensuremath≪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 \ensuremathα\ensuremath{\alpha}\ensuremathα 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 \ensuremathβ\ensuremath≪1\ensuremath{\beta}\ensuremath{\ll}1\ensuremathβ\ensuremath≪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 \ensuremathδ\ensuremath{\delta}\ensuremathδ-function interaction are considered to examine some of the interesting features of the uniform-limit procedure.
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.
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.
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.
The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function g(r)g(\mathcal{r})g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in $\ensuremath{\alpha}=1\ensuremath{-}g(0)$ for all physical quantities which depend on the particle density. The nnn-particle distribution functions p(n){p}^{(n)}p(n) are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the g(r)g(\mathcal{r})g(r) function for n=3and4n=3 \mathrm{and} 4n=3and4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using p(3){p}^{(3)}p(3) function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function U(r)\mathcal{U}(\mathcal{r})U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for U(r)\mathcal{U}(\mathcal{r})U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of U(r)\mathcal{U}(\mathcal{r})U(r), p(3){p}^{(3)}p(3), p(4){p}^{(4)}p(4), and the ground-state energy are computed using the Gaussian approximation for g(r)g(\mathcal{r})g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of p(n){p}^{(n)}p(n) in terms of g(r)g(\mathcal{r})g(r).
The maximum deviation of the radial distribution function g(r)g(r)g(r) from its asymptotic value, $g(\ensuremath{\infty})=1$, is used as an expansion parameter in calculating the properties of a degenerate boson system. The uniform limit, defined by the condition $1\ensuremath{-}g(0)\ensuremath{\ll}1$, holds at low densities under appropriate constraints on the Fourier transforms of the interaction potential and also at high densities for the charged-boson system. In the uniform limit a procedure based on (1) a Jastrow-type trial function, (2) the Wu-Feenberg functional for the kinetic energy, (3) the Kirkwood superposition approximation, or the more accurate Abe form for the three-particle distribution function, yields the Bogoliubov formulas for the ground-state energy and the excitation energies of a degenerate or nearly degenerate boson system. Results for the ground state of the high-density charged-boson system include $\frac{〈\mathrm{KE}〉}{N}=\ensuremath{-}\frac{1}{5}\frac{〈\mathrm{PE}〉}{N}+O({{r}_{s}}^{0}),$ $\frac{E}{N}\ensuremath{\leqq}\ensuremath{-}\frac{0.8031}{{{r}_{s}}^{\frac{3}{4}}}+0.028.$ Numerical results are also computed at intermediate and moderately low densities ($0.01\ensuremath{\leqq}{r}_{s}\ensuremath{\leqq}100$).