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Physical Review

Series Expansion in the Variational Approach to Many-Boson Problems

Deok Kyo Lee

Published Nov 5, 1969• 33 citations
Publisher

Abstract

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 $g(\mathcal{r})$ 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 $n$-particle distribution functions ${p}^{(n)}$ are evaluated to order ${\ensuremath{\alpha}}^{4}$ as functionals in the $g(\mathcal{r})$ function for $n=3 \mathrm{and} 4$ 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 ${p}^{(3)}$ function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order ${\ensuremath{\alpha}}^{4}$, for the two-particle correlation function $\mathcal{U}(\mathcal{r})$, and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for $\mathcal{U}(\mathcal{r})$ yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of $\mathcal{U}(\mathcal{r})$, ${p}^{(3)}$, ${p}^{(4)}$, and the ground-state energy are computed using the Gaussian approximation for $g(\mathcal{r})$, 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 ${p}^{(n)}$ in terms of $g(\mathcal{r})$.

DOI
10.1103/physrev.187.326

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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 \ensuremathα=1\ensuremath−g(0)\ensuremath{\alpha}=1\ensuremath{-}g(0)\ensuremathα=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 \ensuremathα4{\ensuremath{\alpha}}^{4}\ensuremathα4 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 \ensuremathα4{\ensuremath{\alpha}}^{4}\ensuremathα4, 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 ($0.01\ensuremath{\leqq}{r}_{s}\ensuremath{\leqq}100$).
The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.
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 ground state of a one-dimensional system of many bosons interacting through a repulsive $\ensuremath{\delta}$-function potential is studied when the coupling parameter $\ensuremath{\gamma}$ of the system is small. The procedure is based on the variational analysis in the Bijl-Dingle-Jastrow wave-function space. The ground-state energy, determined in a series form in powers of ${\ensuremath{\gamma}}^{\frac{1}{2}}$ and truncated after the third leading term, is found to be in fair agreement with exact results of Lieb and Liniger over a range of the coupling parameter extending up to $\ensuremath{\gamma}=6$. Two slightly different approaches are considered, and they are shown to be equivalent.
The energy and structure of a dilute hard-disks Bose gas are studied in the framework of a variational many-body approach based on a Jastrow correlated ground-state wave function. The asymptotic behaviors of the radial distribution function and the one-body density matrix are analyzed after solving the Euler equation obtained by a free minimization of the hypernetted chain energy functional. Our results show important deviations from those of the available low density expansions, already at gas parameter values $x\ensuremath{\sim}0.001$. The condensate fraction in 2D is also computed and found generally lower than the 3D one at the same xxx.