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Physical Review A

Correlation Energy of an Electron Gas in the Quantum Strong-Field Limit

Gerd E. Keiser, Fa Yueh Wu

Published Oct 1, 1973• 9 citations
Publisher

Abstract

The ground state of an electron gas in an intense magnetic field is studied using a wave function of the product form ${\ensuremath{\Psi}}_{0}^{B}\ensuremath{\Phi}$. The correlation factor ${\ensuremath{\Psi}}_{0}^{B}$ is taken to be the ground-state solution of a charged Bose gas and $\ensuremath{\Phi}$ a determinant of single-particle Landau states. In the quantum strong-field limit so that only the lowest Landau state is populated, the correlation energy is computed using the cluster-expansion technique and a variational determination of the boson energy. Numerical results obtained are lower than those derived under the random-phase approximation.

DOI
10.1103/physreva.8.2094

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Proceedings of the Physical Society•Nov 1, 1962•53 citations
The ground state of an electron gas in an intense magnetic field is studied using a wave function of the product form \ensuremathΨ0B\ensuremathΦ{\ensuremath{\Psi}}_{0}^{B}\ensuremath{\Phi}\ensuremathΨ0B​\ensuremathΦ. The correlation factor \ensuremathΨ0B{\ensuremath{\Psi}}_{0}^{B}\ensuremathΨ0B​ is taken to be the ground-state solution of a charged Bose gas and \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ a determinant of single-particle Landau states. In the quantum strong-field limit so that only the lowest Landau state is populated, the correlation energy is computed using the cluster-expansion technique and a variational determination of the boson energy. Numerical results obtained are lower than those derived under the random-phase approximation.
The ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.
The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where u0{u}_{0}u0​ is the ground-state energy per particle in Rydbergs and rs{r}_{s}rs​ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to rs=10{r}_{s}=10rs​=10. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.
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.
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 ground-state wave function for a quantum electron gas is approximated by $\ensuremath{\psi}=\ensuremath{\Phi}D$, where DDD is the ground-state ideal-gas wave function and $\ensuremath{\Phi}=\mathrm{exp}[\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{i<j}^{}u({r}_{\mathrm{ij}})]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to u(r)u(r)u(r). The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\ensuremath{\sim}(\frac{e}{\ensuremath{\hbar}})\frac{{(\frac{m}{\ensuremath{\pi}\ensuremath{\rho}})}^{\frac{1}{2}}}{r}$.
In a previous paper a trial wave function of the form ψ = D II i<i f ( x ij ) was used variationally to calculate the ground-state energy of an interacting electron gas, D being a determinant of plane waves and f ( x ij ) a correlation function. In this work attention is focused on the nature of the wave function; in particular, the correlation function f ( x ij ) and the pair distribution function are calculated and numerical results are presented for two widely different densities. In the course of this work, further light is thrown on the approximations made in the earlier energy calculations. Finally, the use of a wave function of the type ψ = II i<j f ( x ij ) in the calculation of the ground-state energy of a system of charged bosons is discussed and the exact high-density result of Foldy obtained.