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Physical review. B, Condensed matter

Structure of the ground-state wave function of quantum fluids and "exact" numerical methods

L. Reatto

Published Jul 1, 1982• 18 citations
Publisher

Abstract

The problem of how to extract information on the structure of the exact ground-state wave function of a Bose fluid from the results of Green's-function Monte Carlo computations is considered. Use of the criterion of maximum overlap of the exact ground state with an extended Jastrow function leads to certain equalities for correlation functions. By using methods borrowed from theories of classical fluids various schemes are proposed that permit the determination of the "best" Jastrow function. The case of Fermi statistics is also considered.

DOI
10.1103/physrevb.26.130

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The problem of how to extract information on the structure of the exact ground-state wave function of a Bose fluid from the results of Green's-function Monte Carlo computations is considered. Use of the criterion of maximum overlap of the exact ground state with an extended Jastrow function leads to certain equalities for correlation functions. By using methods borrowed from theories of classical fluids various schemes are proposed that permit the determination of the "best" Jastrow function. The case of Fermi statistics is also considered.
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.
The ground state of a two-dimensional charged quantum system is studied using the method of correlated-basis functions. For particles obeying the Bose-Einstein statistics we compute the ground-state energy variationally for all particle densities using the Bijl-Dingle-Jastrow trial wave function. The Bose ground-state energy and wave function so determined are used next to study the Fermi system. We compute the Fermi ground-state energy using a cluster expansion approach, which is again valid for all particle densities, and find that the ground state is paramagnetic. While these calculations provide variational upper bounds to the ground-state energies, we also establish independent lower bounds to the ground-state energies.
An elaborate cluster analysis of the pair occupancy ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ is performed for a quantum fluid described by a Jastrow wave function. Quantity ${P}_{\stackrel{^}{q}\stackrel{^}{p}}$ measures the simultaneous occupation of single-particle orbitals $\stackrel{^}{q}$, $\stackrel{^}{p}$ and provides information on the presence (or absence) of strongly correlated pairs in the ground state of an extended system of interacting particles. A diagrammatic formulation rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured and demonstrated to convincingly high cluster order that the pair occupancy of Bose fluids contains a finite and positive anomalous portion ${\ensuremath{\chi}}_{q}^{2}$ for pairs of particles with equal and opposite momenta, $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}+\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}=0$, in contrast to a normal Fermi fluid. The leading diagrams necessary for a structural exploration of the pairing function $\ensuremath{\chi}(r)$ which is defined as the Fourier inverse of quantity ${\ensuremath{\chi}}_{q}$ are displayed. It is further demonstrated that a close kinship exists between the function $\ensuremath{\chi}(r)$ and the one-particle density matrix n(r)n(r)n(r). It is also shown that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlation factor implies a singular behavior $\ensuremath{\sim}{q}^{\ensuremath{-}1}$ of the function ${\ensuremath{\chi}}_{q}$ for small momenta $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$. The pairing function $\ensuremath{\chi}(r)$ is calculated numerically for the ground state of liquid 4He^{4}\mathrm{He}4He at equilibrium density adopting the familiar short-ranged correlation factor f(r)f(r)f(r) employed by Schiff and Verlet and others.
"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."
The ground-state energies and pairing gaps in dilute superfluid Fermi gases have now been calculated with the quantum Monte Carlo method without detailed knowledge of their wave functions. However, such knowledge is essential to predict other properties of these gases such as density matrices and pair distribution functions. We present a new and simple method to optimize the wave functions of quantum fluids using the Green's function Monte Carlo method. It is used to calculate the pair distribution functions and potential energies of Fermi gases over the entire regime from atomic Bardeen-Cooper-Schrieffer superfluid to molecular Bose-Einstein condensation, spanned as the interaction strength is varied.