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Physical Review

Charged Boson Gas

Leslie L. Foldy

Published Nov 1, 1961• 217 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.124.649

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arXiv:math-ph/0204027•Apr 11, 2002•5 citations
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.
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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.
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The Brandow boson linked-cluster expansion has been used to redo the Foldy calculation of the Bogoliubov theory for a charged boson gas. The particle-number conservation is treated exactly. The ring diagrams have been summed using the same techniques as used by Gell-Mann and Brueckner for the electron gas. The next-most-divergent diagrams (twisted rings) have also been summed. These diagrams contain a renormalized hole line and thus show explicitly the dynamics of the condensate which is treated only in an average way in the Bogoliubov theory. The results for the ground-state energy and the condensation fraction agree exactly with Foldy in the high-density limit.
Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρρρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m2π\hbar^2 a ρ/m2πℏ2aρ/m where aaa is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5N^{7/5}N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.