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. B, Solid state

Dynamic structure function of a Bose gas at T=0

David L. Bartley, Victor K. Wong

Published Nov 1, 1975• 11 citations
Publisher

Abstract

The dynamic structure function $\mathcal{S}(k, \ensuremath{\omega})$, as a function of wave vector $k$ and frequency $\ensuremath{\omega}$, is calculated rigorously for a simple Bose gas at $T=0$, The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of $\mathcal{S}(k, \ensuremath{\omega})$ valid for arbitrary $k$ and $\ensuremath{\omega}$. Qualitatively, $\mathcal{S}(k, \ensuremath{\omega})$ at finite but small $k$ is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of $\mathcal{S}(k, \ensuremath{\omega})$, the imaginary part of the phonon spectrum, and $\mathcal{S}(k, \ensuremath{\omega})$ itself. Finally, the impulse approximation and the extraction of the condensate density ${n}_{0}$ from $\mathcal{S}(k, \ensuremath{\omega})$ in the large-$k$ limit are discussed.

DOI
10.1103/physrevb.12.3775

Related papers

Theory of a Charged Bose Gas. I

Shang-keng Ma, Chia-Wei Woo

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.

Physical Review•Jul 5, 1967•70 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

Phonons and the Properties of a Bose System

L. Reatto, Geoffrey V. Chester

In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where $\mathrm{ln}f(r)$ has an infinite range. The other factor is assumed to have this same form but with $\mathrm{ln}f(r)$ of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors $k$, the structure factor $S(k)$ is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature $T$, $S(k)$ tends to the constant value $\frac{{k}_{B}T}{m{c}^{2}}$, where $m$ is the mass of the particles and $c$ the velocity of propagation of the phonons. The momentum distribution ${n}_{k}$ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-$k$ behavior of both $S(k)$ and ${n}_{k}$ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.

Physical Review•Mar 5, 1967•281 citations

Many-body Bose systems and the hard-sphere model: dynamic properties from the weak to the strong interaction regime

R. Rota, F. Tramonto, Davide Emilio Galli +1 more

We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.

Journal of Physics Conference Series•Aug 14, 2014•2 citations

Goldstone phonons in a Bose-condensed gas at finite temperature: One-loop approximation

Stephen H. Payne, Adam Griffin

Using a field-theoretic formalism in terms of regular (proper, irreducible) quantities, we give a detailed analysis of the long-wavelength collisionless excitations of a weakly interacting dilute Bose-condensed gas in the intermediate-temperature region. Earlier work by Sz\'epfalusy and Kondor has shown that use of the free-Bose-gas approximation for the regular quantities leads to a phonon with a velocity equal to [${n}_{0}$(T)${V}_{0}$/m${]}^{1/2}$, where ${n}_{0}$(T) is the temperature-dependent condensate of a free Bose gas. In this paper the phonon velocity is worked out with use of the one-loop diagrams for the regular quantities. The result is [${n}_{S}$(T)${V}_{0}$/m${]}^{1/2}$[1-f(T)], where ${n}_{S}$(T) is the Landau superfluid density defined in terms of the Bogoliubov excitations and f(T) is a relatively small correction. Numerical calculations are given for spin-polarized ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ with density n=5\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{3}$. .AE

Physical review. B, Condensed matter•Dec 1, 1985•26 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
The dynamic structure function S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω), as a function of wave vector kkk and frequency \ensuremathω\ensuremath{\omega}\ensuremathω, is calculated rigorously for a simple Bose gas at T=0T=0T=0, The dielectric formulation is used to satisfy general symmetry requirements for a Bose system, and the calculation is carried out in the first approximation beyond Bogoliubov taking into account the three-phonon process. An analytic solution in terms of elliptic integrals is obtained for the width of S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω) valid for arbitrary kkk and \ensuremathω\ensuremath{\omega}\ensuremathω. Qualitatively, S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω) at finite but small kkk is found to have a square-root behavior near the threshold frequency for two-phonon production, a peak at the elementary excitation frequency, and a long power-law tail at high frequencies. Illustrative numerical results are presented for the width of S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω), the imaginary part of the phonon spectrum, and S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω) itself. Finally, the impulse approximation and the extraction of the condensate density n0{n}_{0}n0​ from S(k,\ensuremathω)\mathcal{S}(k, \ensuremath{\omega})S(k,\ensuremathω) in the large-kkk limit are discussed.
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 ${\ensuremath{\omega}}_{\mathrm{pl}}$ (the next order beyond the Bogoliubov approximation). rs{r}_{s}rs​ is the ratio of the average interparticle distance to the Bohr radius and ${\ensuremath{\omega}}_{\mathrm{pl}}$ is the plasma frequency.
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.
In this paper we suggest a new ground-state wave function and low-temperature density matrix for a strongly interacting system of bosons. Our basic assumption is that the system can support long-wavelength phonons and that these can propagate independently of any other mode of motion. We therefore write the ground-state function as the product of two factors. One factor arises from the zero-point motion of the phonons, and we show that it has the form ${\ensuremath{\Pi}}_{i<j}f({r}_{\mathrm{ij}})$, where lnf(r)\mathrm{ln}f(r)lnf(r) has an infinite range. The other factor is assumed to have this same form but with lnf(r)\mathrm{ln}f(r)lnf(r) of finite range; it takes into account the short-range correlations arising from the strong repulsive part of the interparticle potential. The function we have chosen to represent the short-range correlations is not new; functions of this kind were first introduced by Bijl and later by Jastrow. At finite temperatures we use a density matrix for an ensemble of excited phonon states. We find that for small wave vectors kkk, the structure factor S(k)S(k)S(k) is equal to $\frac{\ensuremath{\hbar}k}{2mc}$, a result that was first derived by Feynman. At a finite temperature TTT, S(k)S(k)S(k) tends to the constant value kBTmc2\frac{{k}_{B}T}{m{c}^{2}}mc2kB​T​, where mmm is the mass of the particles and ccc the velocity of propagation of the phonons. The momentum distribution nk{n}_{k}nk​ has a ${k}^{\ensuremath{-}1}$ singularity at absolute zero and a stronger, ${k}^{\ensuremath{-}2}$ singularity at finite temperatures. The small-kkk behavior of both S(k)S(k)S(k) and nk{n}_{k}nk​ is completely controlled by the correlations introduced by the phonons. Both the ground-state function and the density matrix imply that there is a finite fraction of particles in the zero-momentum state in three dimensions; this fraction does not seem to be appreciably affected by the infinite-range correlations introduced by the phonons. We find, however, that these correlations imply that a one-dimensional Bose system does not exhibit Bose-Einstein condensation at any temperature, while a two-dimensional system exhibits Bose-Einstein condensation only at absolute zero.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.
Using a field-theoretic formalism in terms of regular (proper, irreducible) quantities, we give a detailed analysis of the long-wavelength collisionless excitations of a weakly interacting dilute Bose-condensed gas in the intermediate-temperature region. Earlier work by Sz\'epfalusy and Kondor has shown that use of the free-Bose-gas approximation for the regular quantities leads to a phonon with a velocity equal to [n0{n}_{0}n0​(T)V0{V}_{0}V0​/m]1/2{]}^{1/2}]1/2, where n0{n}_{0}n0​(T) is the temperature-dependent condensate of a free Bose gas. In this paper the phonon velocity is worked out with use of the one-loop diagrams for the regular quantities. The result is [nS{n}_{S}nS​(T)V0{V}_{0}V0​/m]1/2{]}^{1/2}]1/2[1-f(T)], where nS{n}_{S}nS​(T) is the Landau superfluid density defined in terms of the Bogoliubov excitations and f(T) is a relatively small correction. Numerical calculations are given for spin-polarized ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ with density n=5\ifmmode\times\else\texttimes\fi{}1018{10}^{18}1018 cm3{\mathrm{cm}}^{3}cm3. .AE
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.