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

Theory of a Charged Bose Gas. I

Shang-keng Ma, Chia-Wei Woo

Published Jul 5, 1967• 70 citations
Publisher

Abstract

This is the first of two papers in which the high-density charged Bose gas at zero temperature is treated by two independent and self-contained methods. In this paper, we are concerned with the excitation spectrum of the system. An analysis of the formal and physical aspects of the theory is followed by a simple numerical calculation. We apply the usual field-theoretic formulation for a Bose system. The Green's functions and other response functions, or propagators, are analyzed, emphasizing the role of the dielectric constant. The general features of the excitation spectrum are described. To apply and to further illustrate the formalism, the low-lying levels are numerically determined to the order of ${{r}_{s}}^{\frac{3}{4}}$ in units of ${\ensuremath{\omega}}_{\mathrm{pl}}$ (the next order beyond the Bogoliubov approximation). ${r}_{s}$ is the ratio of the average interparticle distance to the Bohr radius and ${\ensuremath{\omega}}_{\mathrm{pl}}$ is the plasma frequency.

DOI
10.1103/physrev.159.165

Related papers

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 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

Excitations in a Bose Gas at Finite Temperatures. II. Relation between Single-Particle and Density Fluctuations

Allan Griffin, T. H. Cheung

By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate ${n}_{0}=0$, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of ${n}_{0}$, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary $t$-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.

Physical Review A•Jun 1, 1973•36 citations

Charged Bose gas in two dimensions. Zero magnetic field

D. F. Hines, Norman E. Frankel

The dielectric response of the two-dimensional charged Bose gas is investigated in the random-phase approximation. Using this temperature-dependent dielectric function we find the energy spectrum and the damping for the quasiparticles. The electrostatic potential around a test charge at $T=0$ is displayed. We also study the thermodynamic functions of the gas at very low temperatures and high densities.

Physical review. B, Condensed matter•Aug 1, 1979•41 citations

Dielectric formalism for a Bose fluid

Allan Griffin

In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function G αβ ( Q , ω) appearing as a distinct component of the the dynamic structure factor S ( Q , ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S ( Q , ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between G αβ ( Q , ω) and S ( Q , ω) in a Bose-condensed fluid ( n 0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through T λ, although these excitations have very temperature-dependent weights in G αβ ( Q , ω) and S ( Q , ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about G αβ ( Q , ω) from inelastic neutron-scattering data on S ( Q , ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid. After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4 He.

Cambridge University Press eBooks•Aug 19, 1993•0 citations

Charged Bose Gas

Gideon Carmi, A. J. Lock

By using the Bohm-Pines and Bogoliubov formalisms as well as the random-phase approximation, the investigation of the charged Bose gas is extended into the intermediate densities for which ${r}_{s}\ensuremath{\lesssim}O(1)$. The ground state energy is essentially correlation energy, and is therefore negative as expected. The expression for the plasmon spectrum does possess an energy gap; the single-particle excitation spectrum does not. The specific-heat function is examined to first order in the temperature and is shown to approach zero from above as the temperature does the same. Finally, the expression for the pressure is shown to persist in remaining negative for all ${r}_{s}\ensuremath{\lesssim}O(1)$ at zero temperature.

Physical Review A•Mar 1, 1972•3 citations
This is the first of two papers in which the high-density charged Bose gas at zero temperature is treated by two independent and self-contained methods. In this paper, we are concerned with the excitation spectrum of the system. An analysis of the formal and physical aspects of the theory is followed by a simple numerical calculation. We apply the usual field-theoretic formulation for a Bose system. The Green's functions and other response functions, or propagators, are analyzed, emphasizing the role of the dielectric constant. The general features of the excitation spectrum are described. To apply and to further illustrate the formalism, the low-lying levels are numerically determined to the order of rs34{{r}_{s}}^{\frac{3}{4}}rs​43​ in units of \ensuremathωpl{\ensuremath{\omega}}_{\mathrm{pl}}\ensuremathωpl​ (the next order beyond the Bogoliubov approximation). rs{r}_{s}rs​ is the ratio of the average interparticle distance to the Bohr radius and \ensuremathωpl{\ensuremath{\omega}}_{\mathrm{pl}}\ensuremathωpl​ is the plasma frequency.
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 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.
By making a systematic analysis of correlation functions in terms of irreducible and reducible parts, Ma and others have developed a dielectric formulation of the dynamics of a condensed Bose system which fully includes the background (or excited) atoms. We show that this formulation quite generally leads to the density and single-particle-correlation functions both having two resonances ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ although with different weights. If the condensate n0=0{n}_{0}=0n0​=0, the ${\ensuremath{\omega}}_{2}$ mode only appears in the single-particle spectrum while ${\ensuremath{\omega}}_{1}$ only appears in the density-fluctuation spectrum. For finite values of n0{n}_{0}n0​, these modes are coupled and renormalized to ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$. As a specific illustration, we use the shielded-potential approximation (SPA) for the reducible self-energies. In this model, we find the free-particle excitations (${\ensuremath{\omega}}_{2}$) are coupled to the zero-sound density fluctuations (${\ensuremath{\omega}}_{1}$) through the action of the condensate. In the SPA, the reducible self-energies have a pole at ${\ensuremath{\omega}}_{1}$. The usual Bogoliubov, Hartree-Fock, and ordinary ttt-matrix approximations involve nonsingular self-energies and hence do not exhibit the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ mode. Experimentally, it appears that the high-frequency ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ mode has never been detected, but this is probably owing to the fact that it is strongly damped as a result of its coupling to the zero-sound mode. Using the two-fluid hydrodynamic equations, one can argue that at low frequencies the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{1}$ excitation corresponds to first sound and the ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{2}$ excitation corresponds to second sound. Finally, we briefly discuss the possibility of singularities in the irreducible self-energies at high frequencies and the resulting two-roton bound states.
The dielectric response of the two-dimensional charged Bose gas is investigated in the random-phase approximation. Using this temperature-dependent dielectric function we find the energy spectrum and the damping for the quasiparticles. The electrostatic potential around a test charge at T=0T=0T=0 is displayed. We also study the thermodynamic functions of the gas at very low temperatures and high densities.
In Chapter 3, we have shown how a Bose condensate leads to the single-particle Green's function G αβ ( Q , ω) appearing as a distinct component of the the dynamic structure factor S ( Q , ω). This feature is a direct consequence of (3.13) and is explicitly shown by the expression for S ( Q , ω) in (3.18). In the present chapter, we review the many-body diagrammatic analysis which specifies the precise relation between G αβ ( Q , ω) and S ( Q , ω) in a Bose-condensed fluid ( n 0 ≠ 0). The power of this “dielectric formalism” lies in showing that both functions mirror each other, as a result of the condensate-induced intermixing of single-particle and density excitations. At the same time, the formalism emphasizes the continuity of these excitations as the fluid passes through T λ, although these excitations have very temperature-dependent weights in G αβ ( Q , ω) and S ( Q , ω). While it is the poles of the single-particle Green's function which describe the elementary excitations, experimentally it is difficult to probe directly this field-fluctuation spectrum. In particular, it is only in a Bose-condensed fluid that one can obtain information about G αβ ( Q , ω) from inelastic neutron-scattering data on S ( Q , ω). In the absence of a condensate, there is little relation between the density-fluctuation modes and the single-particle excitations in a Bose fluid. After setting up the dielectric formalism in Section 5.1, the next two sections consider some model approximations at finite temperatures. These models are most appropriate to a dilute weakly interacting Bose gas (WIDBG), but we believe they also give crucial insight into various aspects of superfluid 4 He.
By using the Bohm-Pines and Bogoliubov formalisms as well as the random-phase approximation, the investigation of the charged Bose gas is extended into the intermediate densities for which ${r}_{s}\ensuremath{\lesssim}O(1)$. The ground state energy is essentially correlation energy, and is therefore negative as expected. The expression for the plasmon spectrum does possess an energy gap; the single-particle excitation spectrum does not. The specific-heat function is examined to first order in the temperature and is shown to approach zero from above as the temperature does the same. Finally, the expression for the pressure is shown to persist in remaining negative for all ${r}_{s}\ensuremath{\lesssim}O(1)$ at zero temperature.