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Physical Review A

Diagrammatic Analysis of the Method of Correlated Basis Functions. II. Perturbation Theory for a Weakly Interacting Bose Gas

Hock-Kee Sim, Chia-Wei Woo

Published Nov 1, 1970• 22 citations
Publisher

Abstract

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 $O({\ensuremath{\lambda}}^{5})$.

DOI
10.1103/physreva.2.2032

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Diagrammatic Analysis of the Method of Correlated Basis Functions. I. Jastrow Correlating Factor for a Weakly Interacting Bose Gas

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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(\ensuremathλ4)O({\ensuremath{\lambda}}^{4})O(\ensuremathλ4). It is also demonstrated indirectly that the Jastrow function sums rings and ladders at least up to O(\ensuremathλ5)O({\ensuremath{\lambda}}^{5})O(\ensuremathλ5).
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.
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.
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.
We study the connections between the chain summations of variational theories of Fermi fluids and those generated by ring-diagram summations in the correlated-basis-functions (CBF) perturbation theory. The effective interactions of CBF theory and the random-phase-approximation (RPA) equations in the correlated basis are rewritten in an irreducible form. We discuss local approximations to the irreducible vertex which are sympathetic with the use of variational wave functions. They allow contact to be made with conventional formulations of the RPA, to weak-coupling, pseudopotential, or "local-screening" theories. Our studies lead also to consistency requirements between variational calculations of ground-state properties and the effective interactions used in CBF theory.
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.