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Physical Review Letters

Diagrammatic Analysis of the Method of Correlated Basis Functions

Hock-Kee Sim, Chia-Wei Woo, J. R. Buchler

Published May 18, 1970• 13 citations
Publisher

Abstract

The weakly interacting Bose gas is studied on the one hand with the independent-particle perturbation formalism of Hugenholtz and Pines, and on the other hand by a perturbation theory using a set of correlated basis functions. We find by comparison that up to fourth order in energy, a Jastrow correlated wave function sums all rings and ladders, and other diagrams in part; and the simplest correlated basis functions perturbation correction accounts for all remaining discrepancies.

DOI
10.1103/physrevlett.24.1094

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Diagrammatic Analysis of the Method of Correlated Basis Functions. II. Perturbation Theory 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 $O({\ensuremath{\lambda}}^{4})$. It is also demonstrated indirectly that the Jastrow function sums rings and ladders at least up to $O({\ensuremath{\lambda}}^{5})$.

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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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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.

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arXiv:2303.01124•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2023•0 citations
The weakly interacting Bose gas is studied on the one hand with the independent-particle perturbation formalism of Hugenholtz and Pines, and on the other hand by a perturbation theory using a set of correlated basis functions. We find by comparison that up to fourth order in energy, a Jastrow correlated wave function sums all rings and ladders, and other diagrams in part; and the simplest correlated basis functions perturbation correction accounts for all remaining discrepancies.
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 $O({\ensuremath{\lambda}}^{5})$.
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.
A correlated wave function for interacting Bose systems with correlation factors up to five-particle parts is suggested. The vehicle used is an analysis for the weakly interacting Bose gas by a new method. Terms up to $O({\ensuremath{\lambda}}^{2})$, where $\ensuremath{\lambda}$ measures the strength of the interaction, for the two-particle (Jastrow) factor and the three-particle factor, as well as ground-state energy up to $O({\ensuremath{\lambda}}^{4})$, are shown to agree with exact results obtained previously with other methods. New results include all multiparticle correlation factors up to $O({\ensuremath{\lambda}}^{4})$. The wave function will be most useful for analytically soluble model systems, and more importantly, for Monte Carlo or molecular-dynamics calculations dealing with realistic systems such as superfluid 4He^{4}\mathrm{He}4He.
Abstract We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.