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Physical Review Letters

Ground-State Energy of a Charged-Boson Gas

Deok Kyo Lee

Published Apr 12, 1971• 7 citations
Publisher

Abstract

The ground-state energy of a charged-boson gas at high density is re-examined and it is found that Brueckner's exact result for the second-order energy is reproduced precisely by Feenberg's variation-perturbation method.
DOI
10.1103/physrevlett.26.877

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The ground-state energy of a charged boson gas at high density is evaluated exactly to second order in the density expansion, making use of complete sums of one- and two-ring diagrams. The result agrees numerically with Feenberg's variational calculation of the energy, although the methods appear to be substantially different. The presence of logarithmic terms in the energy, which are absent in the exact result, is shown to be a result of truncation of the Hamiltonian in the pair approximation or of inadequacies of variational wave functions which omit interpair correlations.
The simultaneous effects of the condensate and of the long-range nature of the interaction have made the high-density charged Bose gas an interesting theoretical problem. In this paper we present a straight-forward procedure which takes these effects into account. It is based on the perturbation theory in the Bogoliubov representation and the principle of the compensation of dangerous diagrams. Both the ground-state energy and the excitation spectrum are studied. Numerical results are given for the first two orders in rs34{{r}_{s}}^{\frac{3}{4}}rs​43​. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.
The ground state energy and elementary excitations of a charged gas of bosons at high densities are examined by use of the method developed by Bogoliubov for boson gases. It is conjectured, but not herein established, that this method yields exact results in the high-density limit analogous to those obtained by Gell-Mann and Brueckner, and by Sawada, in the corresponding case of a charged fermion gas. The ground state energy is essentially correlation energy, and is therefore negative, and its magnitude varies as the one-fourth power of the density at high densities. The elementary excitations have for low momenta the energy appropriate to plasma waves, and for high momenta the energy appropriate to single-particle excitation. There is therefore an energy gap, suggesting that the gas is both a superfluid and a superconductor at low temperatures. At low densities the behavior of a charged gas is independent of statistics; hence, such a gap must disappear as the system is expanded.
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 ${r}_{s}\ensuremath{\sim}5$.
Foldy, and later Girardeau, calculated the ground-state energy of a charged Bose gas at high density. We rederive the common first term obtained by these authors by using a nonperturbation method developed previously. Our aims are: (i) to establish the validity of this common result, which has not been proved; (ii) to establish the validity and usefulness of our nonperturbation method. We also show that our method will give the correct functional dependence of the ground-state energy on the density at low density, although the exact coefficient must await a numerical computation.