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Physical Review

Ground-State Energy of Bose-Einstein Gas with Repulsive Interaction

K. Sawada

Published Dec 15, 1959• 63 citations
Publisher

Abstract

The explicit evaluation of the ground-state energy of a Bose gas interacting through a two-body repulsive potential as an expansion essentially in powers of the density of the particles and the scattering length is given. It is shown that the problem can be treated by using diagrammatical analysis up to any orders. The energy was investigated up to the fourth order in the ordinary sense of a perturbation expansion in terms of the scattering length, but including terms up to infinite order for special classes of diagrams. The energy obtained contains in general a $\ensuremath{\Sigma}{n}^{}{C}_{n}{\ensuremath{\rho}}^{n}\mathrm{ln}\ensuremath{\rho}$ ($n\ensuremath{\geqq}2$, half-integer and integer) dependence, and the coefficient ${C}_{2}$ is evaluated. A self-consistent treatment is given following this analysis and is shown to lead to the formulation of Beliaev.

DOI
10.1103/physrev.116.1344

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Bose-Einstein Gas with Repulsive Interactions: General Theory

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The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and $a$ the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach $\frac{{q}^{2}}{2M}$ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.

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The Ground State Energy of a Dilute Bose Gas in Dimension n >3

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We consider a Bose gas in spatial dimension $n>3$ with a repulsive, radially symmetric two-body potential $V$. In the limit of low density $ρ$, the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}ρ$, where $|\mathbb S^{n-1}|$ denotes the surface measure of the unit sphere in $\mathbb{R}^n$ and $a$ is the scattering length of $V$. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term $(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|$ in dimension $n=4$, where $γ:=\int V(x)|x|^{-2}\, dx$, and a correction term which is $\mathcal{O}(ρ^2)$ in higher dimensions.

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Ground-State Energy and Excitation Spectrum of a System of Interacting Bosons

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In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy $E(\mathrm{k})$ of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where $n$ is the density and ${f}_{0}$ is the scattering length for the assumed two-body interaction between the bosons. The coefficients $a$ and $b$ are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient $b$ is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.

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EFFECT OF HIGHER ORDER ENERGY CORRECTIONS INCLUDING THREE-BODY INTERACTION ON THE BOSE-EINSTEIN CONDENSATE WITH THE VARIATION OF REPULSIVE SELF INTERACTION ENERGY

Shri Prakash Tewari, Poonam Silotia, Aditya Saxena +1 more

Various ground state properties such as chemical potential, differential and total energy per particle etc. of Bose-Einstein condensate of 10000 85 Rb atoms with varying repulsive self-interaction energy have been reported considering not only two-body interaction but also including the higher order terms of the low-density energy expansion of homogeneous Bose gas in the Ginzburg, Pitaevskii and Gross (GPG) equation. These include hard-core approximation of the bosons neglected earlier and the three-body interaction terms. The study is more general as it includes the terms beyond logarithm in energy density expansion. It is also shown that such a consideration does not violate the lower bound predicted earlier in which the 'constant' beyond logarithm term in the three-body interaction was neglected.

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Simplified Approach to the Ground-State Energy of an Imperfect Bose Gas

Élliott H. Lieb

The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where $a$ is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.

Physical Review•Jun 15, 1963•106 citations
The explicit evaluation of the ground-state energy of a Bose gas interacting through a two-body repulsive potential as an expansion essentially in powers of the density of the particles and the scattering length is given. It is shown that the problem can be treated by using diagrammatical analysis up to any orders. The energy was investigated up to the fourth order in the ordinary sense of a perturbation expansion in terms of the scattering length, but including terms up to infinite order for special classes of diagrams. The energy obtained contains in general a \ensuremathΣnCn\ensuremathρnln\ensuremathρ\ensuremath{\Sigma}{n}^{}{C}_{n}{\ensuremath{\rho}}^{n}\mathrm{ln}\ensuremath{\rho}\ensuremathΣnCn​\ensuremathρnln\ensuremathρ (n\ensuremath≧2n\ensuremath{\geqq}2n\ensuremath≧2, half-integer and integer) dependence, and the coefficient C2{C}_{2}C2​ is evaluated. A self-consistent treatment is given following this analysis and is shown to lead to the formulation of Beliaev.
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
The ground-state energy per particle E/NE/NE/N and condensate density n0n_0n0​ of a dilute Bose gas are studied with a self-consistent perturbation expansion satisfying the Hugenholtz-Pines theorem and conservation laws simultaneously. A class of Feynman diagrams for the self-energy, which has escaped consideration so far, is shown to add an extra constant cip∼O(1)c_{ip}\sim O(1)cip​∼O(1) to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as E/N=(2πℏ2an/m)[1+(128/15π+16cip/5)a3n]E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]E/N=(2πℏ2an/m)[1+(128/15π​+16cip​/5)a3n​] and n0/n=1−(8/3π+cip)a3nn_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}n0​/n=1−(8/3π​+cip​)a3n​, where aaa, nnn, and mmm are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for cipc_{ip}cip​; the third-order perturbation expansion yields cip=0.412c_{ip}=0.412cip​=0.412.
We consider a Bose gas in spatial dimension n>3n>3n>3 with a repulsive, radially symmetric two-body potential VVV. In the limit of low density ρρρ, the ground state energy per particle in the thermodynamic limit is shown to be (n−2)∣Sn−1∣an−2ρ(n-2)|\mathbb S^{n-1}|a^{n-2}ρ(n−2)∣Sn−1∣an−2ρ, where ∣Sn−1∣|\mathbb S^{n-1}|∣Sn−1∣ denotes the surface measure of the unit sphere in Rn\mathbb{R}^nRn and aaa is the scattering length of VVV. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term (1+Cγ)8π4a6ρ2∣ln⁡(a4ρ)∣(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|(1+Cγ)8π4a6ρ2∣ln(a4ρ)∣ in dimension n=4n=4n=4, where γ:=∫V(x)∣x∣−2 dxγ:=\int V(x)|x|^{-2}\, dxγ:=∫V(x)∣x∣−2dx, and a correction term which is O(ρ2)\mathcal{O}(ρ^2)O(ρ2) in higher dimensions.
In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy E(k)E(\mathrm{k})E(k) of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where nnn is the density and f0{f}_{0}f0​ is the scattering length for the assumed two-body interaction between the bosons. The coefficients aaa and bbb are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient bbb is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
Various ground state properties such as chemical potential, differential and total energy per particle etc. of Bose-Einstein condensate of 10000 85 Rb atoms with varying repulsive self-interaction energy have been reported considering not only two-body interaction but also including the higher order terms of the low-density energy expansion of homogeneous Bose gas in the Ginzburg, Pitaevskii and Gross (GPG) equation. These include hard-core approximation of the bosons neglected earlier and the three-body interaction terms. The study is more general as it includes the terms beyond logarithm in energy density expansion. It is also shown that such a consideration does not violate the lower bound predicted earlier in which the 'constant' beyond logarithm term in the three-body interaction was neglected.
The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where aaa is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.