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Physical review. B, Condensed matter

Ground-state energy of charged quantum fluids in two dimensions

Hock-Kee Sim, R. Tao, Fa Yueh Wu

Published Nov 15, 1986• 27 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.34.7123

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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.
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.
Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at rs=139{r}_{s}=139rs​=139, where rs{r}_{s}rs​ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at rs=2{r}_{s}=2rs​=2, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).
The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.
"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 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.