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Physical Review A

Excitation Spectrum and Liquid-Structure Function of a Charged Bose Gas

A. Bhattacharyya, Chia-Wei Woo

Published Jan 1, 1973• 11 citations
Publisher

Abstract

The excitation spectrum of a charged Bose gas in the high-density limit is calculated to two orders in the expansion parameter ${r}_{s}^{\frac{3}{4}}$ by the method of correlated basis functions (CBF). To $O({r}_{s}^{\frac{3}{4}})$, the result differs from that obtained variationally, but agrees with that obtained by Ma and Woo via diagrammatic perturbation theory. The choice of the basis functions is, however, not unique. By choosing an alternate set of CBF which employs the ground-state eigenfunction as a correlating factor, we find it possible to extract from the excitation spectrum an expression for the liquid-structure function, also to two orders in ${r}_{s}^{\frac{3}{4}}$, which represents a new result. This new facet of CBF helps to demonstrate the flexibility of the method and the ease with which it can be applied to calculating properties of many-particle systems, model as well as realistic.

DOI
10.1103/physreva.7.204

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The excitation spectrum of a charged Bose gas in the high-density limit is calculated to two orders in the expansion parameter rs34{r}_{s}^{\frac{3}{4}}rs43​​ by the method of correlated basis functions (CBF). To O(rs34)O({r}_{s}^{\frac{3}{4}})O(rs43​​), the result differs from that obtained variationally, but agrees with that obtained by Ma and Woo via diagrammatic perturbation theory. The choice of the basis functions is, however, not unique. By choosing an alternate set of CBF which employs the ground-state eigenfunction as a correlating factor, we find it possible to extract from the excitation spectrum an expression for the liquid-structure function, also to two orders in rs34{r}_{s}^{\frac{3}{4}}rs43​​, which represents a new result. This new facet of CBF helps to demonstrate the flexibility of the method and the ease with which it can be applied to calculating properties of many-particle systems, model as well as realistic.
In the continuing effort to establish contacts between the method of correlated basis functions (CBF) and the diagrammatic perturbation theory, we carry out parallel calculations of the excitation spectrum of a weakly interacting Bose gas. The Hugenholtz---Pines (HP) theory is first used to obtain the exact liquid-structure function and the energy spectrum, each to three orders in the strength parameter $\ensuremath{\lambda}$ that characterizes the interaction between a pair of particles in the system. Next we construct sets of CBF using three different correlating factors, and in each case calculate the excitation energy of the system. The results are compared to those obtained from the HP theory. Complete agreement is found to all orders considered. In conjunction with a paper to follow, the analysis demonstrates the flexibility of the CBF method and the relative ease of application in comparison to other methods.
By computing the excitation spectrum of the charged Bose gas, we demonstrate explicitly the distinction between Jastrow's theory and Feenberg's method of correlated basis functions. Using a Jastrow correlating factor in the correlated basis functions, we obtain the spectrum exactly to two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. A rearranged expression results when the true ${\ensuremath{\psi}}_{0}$ is employed as a correlating factor. We take advantage of the difference between these two procedures to obtain S(k)S(k)S(k) exactly to two orders.
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 ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.
Continuing the diagrammatic analysis of the method of correlated basis functions (CBF) for the weakly interacting Bose gas, as reported in the preceding paper, we take off from the unperturbed problem, which is solved essentially by a variational method using a Jastrow correlating factor as the trial wave function. We compute low-order corrections to the unperturbed energy, using a set of correlated basis functions. It is found that the simplest second-order correction in the CBF accounts for all discrepancies (between the Hugenholtz-Pines theory and the unperturbed CBF calculation) which arise in the fourth order of the interaction strength $O({\ensuremath{\lambda}}^{4})$. It is also demonstrated indirectly that the Jastrow function sums rings and ladders at least up to $O({\ensuremath{\lambda}}^{5})$.
As a hypothetical inhomogeneous quantum many-body model, we take a weakly interacting Bose gas made inhomogeneous by the presence of a weak external field. The interaction is assumed to be Fourier transformable and characterized by a strength parameter \ensuremath{\lambda}; this is also true for the external field, characterized by a strength parameter \ensuremath{\varepsilon}. The ground-state energy is calculated to order ${\ensuremath{\varepsilon}}^{2}$${\ensuremath{\lambda}}^{2}$ using three methods: diagrammatic perturbation theory, correlated basis functions (CBF), and density-functional theory (DFT). The diagrammatic perturbation theoretical calculation is new. It differs from our earlier approach for homogeneous Bose systems which evolved from the Hugenholtz-Pines theory. We are able to reproduce the same results in an efficient manner. For the inhomogeneous system, the results are used to measure CBF against DFT. It is found that in their simplest forms CBF yields results which are better than the renormalized DFT by Ebner and Saam. In particular, CBF is superior to DFT in the local-density approximation, even though the latter was designed to take care of systems with slow and small density variations such as the model considered here.