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Physical review. B, Condensed matter

Theory of inhomogeneous quantum systems. I. Static properties of Bose fluids

E. Krotscheck, Guo-Xin Qian, Walter Kohn

Published Apr 1, 1985• 97 citations
Publisher

Abstract

We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of $^{4}\mathrm{He}$ atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.

DOI
10.1103/physrevb.31.4245

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We develop a variational theory for the determination of the ground-state properties of an inhomogeneous Bose fluid at zero temperature. Euler-Lagrange equations are derived for the one- and two-body correlations, and systematic approximation methods are physically motivated. It is shown that the optimized variational method provides a self-consistent approximate summation of ladder and ring diagrams. Numerical applications are presented for the density profile, distribution functions, energy, and chemical potential of films of 4He^{4}\mathrm{He}4He atoms. Both the bulk and surface energies are calculated and found to be in fair agreement with experiments.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."
We study a variational multiparticle correlation theory for the ground state of inhomogeneous Bose fluids at zerotemperature. The ground state wave function of a many-boson system is written as a generalized Jastrow–Feenberg function including multiparticle correlation functions. Euler–Lagrange equations for the one, two, three and four particle correlation functions are derived using a systematic functional theoretical approach within the framework of the generalized hypernetted chain theory (HNC2 equations). The Euler–Lagrange equations for the optimal one and two particle correlation functions are analyzed including the infinite series of the elementary diagrams summed by integral equations. For the determination of the optimal triplet correlation factor we propose a novel iterative procedure in which the HNC2 equation is used to calculate the triplet distribution function.
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.
The variational density-matrix approach introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.
The density functional theory originally developed by Hohenberg, Kohn, and Sham provides a rigorous conceptual framework for dealing with inhomogeneous interacting Fermi systems. We extend this approach to deal with inhomogeneous interacting Bose-condensed systems, limiting this presentation to setting up the formalism to deal with ground-state (T = 0) properties. The key new feature is that one must deal with energy functionals of both the local density n(r) and the local complex macroscopic wave function Φ(r) associated with the Bose broken-symmetry (the local condensate density is [Formula: see text]. Implementing the Kohn–Sham scheme, we reduce the problem to a gas of weakly interacting Bosons moving in self-consistent diagonal and off-diagonal one-body potentials. Our formalism should provide the basis for studies of the surface properties of liquid 4He as well as the properties of Bose-condensed atomic gases trapped in external potentials.
The density functional theory originally developed by Hohenberg, Kohn and Sham provides a rigorous conceptual framework for dealing with inhomogeneous interacting Fermi systems. We extend this approach to deal with inhomogeneous interacting Bose-condensed systems, limiting this presentation to setting up the formalism to deal with ground state (T = 0) properties. The key new feature is that one must deal with energy functionals of both the local density n(r) and the local complex macroscopic wavefunction Φ(r) associated with the Bose broken-symmetry (the local condensate density is nc(r) = |Φ(r) | 2). Implementing the Kohn-Sham scheme, we reduce the problem to a gas of weakly-interacting Bosons moving in self-consistent diagonal and off-diagonal one-body potentials. Our formalism should provide the basis for studies of the surface properties of liquid 4He as well as the properties of Bose-condensed atomic gases trapped in external potentials.