Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Ground State of the Charged Bose Gas

Marvin D. Girardeau

Published Sep 1, 1962• 49 citations
Publisher

Abstract

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 ${u}_{0}$ is the ground-state energy per particle in Rydbergs and ${r}_{s}$ 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 ${r}_{s}=10$. 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 ${r}_{s}\ensuremath{\sim}5$.

DOI
10.1103/physrev.127.1809

Related papers

Ground-State Energy of the Charged Bose Gas

Donald Wright

The ground-state energy of the charged Bose gas at high densities is determined by the method of Green's functions. The pair Hamiltonian of Girardeau and Arnowitt is used to calculate the Green's functions. The leading terms in the high-density expansion are found to agree exactly with those determined by Girardeau using a variational method.

Physical Review•Mar 4, 1966•17 citations

Ground state of the strongly coupled, charged Bose gas

Jean-Pierre Hansen, Redha Mazighi

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 ${r}_{s}=139$, where ${r}_{s}$ 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 ${r}_{s}=2$, 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 (${r}_{s}\ensuremath{\gg}1$).

Physical Review A•Sep 1, 1978•40 citations

Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas. II. Charged Bose Gas at High Density

Élliott H. Lieb, A.Y. Sakakura

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.

Physical Review•Feb 17, 1964•39 citations

Charged Bose gas at T=0

Álvaro A. Caparica, Oscar Hipólito

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.

Physical Review A•Nov 1, 1982•26 citations

Charged Bose Gas

Michael Schick C, Tzong‐Ming Wu

Green-function techniques are applied to the charged Bose gas at the absolute zero of temperature. The procedure permits some general remarks to be made concerning the expansions of the ground-state energy and depletion parameter in terms of ${\mathcal{r}}_{S}$, the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$ as had been conjectured, but additional functions of ${\mathcal{r}}_{S}$ as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$, is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.

Physical Review•Jan 5, 1969•18 citations

Charged Boson Gas at High Density

K. A. Brueckner

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.

Physical Review•Apr 5, 1967•67 citations
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 (rs\ensuremath≪1{r}_{s}\ensuremath{\ll}1rs​\ensuremath≪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 u0=\ensuremath−0.804rs\ensuremath−34\ensuremath−(18)lnrs+O(1){u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)u0​=\ensuremath−0.804rs​\ensuremath−43​\ensuremath−(81​)lnrs​+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 r0\ensuremath∼rs\ensuremath−12\ensuremathρ\ensuremath−13{r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}r0​\ensuremath∼rs​\ensuremath−21​\ensuremathρ\ensuremath−31​, 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 rs\ensuremath∼5{r}_{s}\ensuremath{\sim}5rs​\ensuremath∼5.
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 (${r}_{s}\ensuremath{\gg}1$).
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.
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.
Green-function techniques are applied to the charged Bose gas at the absolute zero of temperature. The procedure permits some general remarks to be made concerning the expansions of the ground-state energy and depletion parameter in terms of rS{\mathcal{r}}_{S}rS​, the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of rS34{{\mathcal{r}}_{S}}^{\frac{3}{4}}rS​43​ as had been conjectured, but additional functions of rS{\mathcal{r}}_{S}rS​ as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of rS34{{\mathcal{r}}_{S}}^{\frac{3}{4}}rS​43​, is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.
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.