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Physical Review A

Effective interactions, linear response, and correlated rings: A study of chain diagrams in correlated basis functions

E. Krotscheck

Published Dec 1, 1982• 70 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.26.3536

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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 description of Fermi systems based on Correlated Basis Functions (CBF) is a natural extension of the variational approach, in which the trial ground state wave function is written in the form. The main problem associated with the neglect, or the approximate treatment, of elementary diagrams is the possible violation of the variational principle, which may become significant at high density. Besides the capability to systematically improve upon the variational estimates of the ground-state energy, the CBF formalism provides a consistent framework to carry out calculations of different nuclear matter properties. In CBF perturbation theory, one has to evaluate matrix elements of the bare nuclear Hamiltonian, the effects of correlations being taken into account by the transformation of the basis states describing the non interacting system. However, the same result can in principle be obtained transforming the Hamiltonian, and using the Fermi gas basis.
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 $O({\ensuremath{\lambda}}^{5})$.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
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.
Correlated-basis-function (CBF) theory, which has provided a firm foundation for ab initio calculations of the properties of quantum fluids such as liquid 4He^{4}\mathrm{He}4He, is adapted and applied at the Jastrow-Feenberg variational level to give an optimized description of the structure and of the elementary excitations of the Ising-spin model in a transverse magnetic field. A set of trial wave functions of Hartree-Jastrow form is assumed to describe the spatial correlations present in the ordered as well as the disordered ground states, and the Feynman Ansatz is consistently adopted to represent the elementary magnon states. The CBF analysis of the spin system employs the hypernetted-chain (HNC) formalism for a substitutional binary mixture of bosons and derives HNC equations for the spatial distribution functions which determine the ground-state energy expectation value. Functional variation of this quantity with respect to the magnetic order parameter and the trial states leads to two Euler-Lagrange equations, which may be interpreted as a renormalized Hartree equation for the optimal magnetization and as a paired-magnon equation for the magnetic correlation function that is analogous to the familiar paired-phonon equation for conventional quantum fluids.Numerical calculations are based on simple cubic lattices and an optimized nearest-neighbor Ansatz for the generating pseudopotential. Results are reported on the order parameter, the energy per lattice site, the transverse magnetization, and the magnon excitation energies as functions of the coupling parameter 0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}\ensuremath{\infty} measuring the strength of the transverse magnetic field. We also present numerical results on the magnetic correlation function, the static structure function, and the correlation length. The system exhibits a second-order phase transition at a critical value ${\ensuremath{\lambda}}_{\mathit{c}}$ of the coupling strength. Our numerical calculations of the optimal order parameter yields ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}5.17 for a simple cubic lattice and ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}3.12 for a square planar lattice. The calculated data are in very good agreement with results derived from perturbation expansions in conjunction with Pad\'e techniques. \textcopyright{} 1996 The American Physical Society.