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Physical Review Letters

Jastrow Theory versus Correlated Basis Functions: Excitation Spectrum of the Charged Bose Gas

Archana Bhattacharyya, Chia-Wei Woo

Published May 15, 1972• 11 citations
Publisher

Abstract

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 ${{r}_{s}}^{\frac{3}{4}}$. 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 $S(k)$ exactly to two orders.

DOI
10.1103/physrevlett.28.1320

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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 \ensuremathψ0{\ensuremath{\psi}}_{0}\ensuremathψ0​ 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 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.
The method of correlated basis functions (CBF) is examined for the special case of a weakly interacting Bose gas. Using the Hugenholtz-Pines theory, we compute the ground-state energy exactly to fourth order in the interaction strength. The resulting expression is compared term by term with the unperturbed ground-state energy computed in the CBF using a Jastrow function as the correlating factor. We find that the three leading orders are all accounted for by the Jastrow function, while beginning in the fourth order only selected terms are included. The use of a correlated wave function in effect corresponds to summations of sellected diagrams to all orders. In particular, the ring diagrams and the ladder diagrams are most susceptible to these summations. In a separate paper we shall examine effects of perturbation in the CBF.
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 $O({\ensuremath{\lambda}}^{5})$.
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.
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 collective-coordinate and correlated-basis-function approaches are systematically applied to the determination of the elementary excitation spectrum $\ensuremath{\omega}(k)$ and the static structure function S(k)S(k)S(k) for two models of a Bose gas which can be characterized by a single small expansion parameter ggg. Results are obtained to first order in ggg, the leading order in which multiexcitation processes are present, and are compared with those found by the microscopic-dielectric-function approach. It is found that a collective-coordinate calculation of $\ensuremath{\omega}(k)$ by Sunakawa, Yamasaki, and Kebukawa is incomplete to first order in ggg and that the convolution approximation for the three-particle distribution function in the correlated-basis-function formalism leads to incorrect results for overlap matrix element and for S(k)S(k)S(k). A form for the overlap matrix element is proposed which leads to correct first-order results. Various properties of $\ensuremath{\omega}(k)$, S(k)S(k)S(k), and the dynamic structure function are discussed.