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arXiv:math-ph/9910033

The Ground State Energy of a Dilute Bose Gas

Élliott H. Lieb, Jakob Yngvason

Published Oct 20, 1999• 9 citations
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Abstract

According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density $ρ$ is $2π\hbar^2ρa/m$ to leading order in $ρa^3\ll 1$, where $a$ is the scattering length of the interaction potential and $m$ the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than $1/r^3$ at infinity.

DOI
10.48550/arxiv.math-ph/9910033

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According to a formula that was put forward many decades ago the ground state energy per particle of an interacting, dilute Bose gas at density ρρρ is 2πℏ2ρa/m2π\hbar^2ρa/m2πℏ2ρa/m to leading order in ρa3≪1ρa^3\ll 1ρa3≪1, where aaa is the scattering length of the interaction potential and mmm the particle mass. This result, which is important for the theoretical description of current experiments on Bose-Einstein condensation, has recently been established rigorously for the first time. We give here an account of the proof that applies to nonnegative, spherically symmetric potentials decreasing faster than 1/r31/r^31/r3 at infinity.
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.
Now that the properties of low temperature Bose gases at low density, $\ensuremath{\rho}$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors four to five decades ago. One of these is that the leading term in the energy/particle is $2\ensuremath{\pi}{\ensuremath{\Elzxh}}^{2}\ensuremath{\rho}a/m$, where aaa is the scattering length. Owing to the delicate and peculiar nature of bosonic correlations, four decades of research have failed to establish this plausible formula rigorously. The only known lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct bound is proved here.
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.
Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density ρ\rho ρ ρ in 2 dimensions in the thermodynamic limit e2D(ρ)=4πρ2Y(1−Y∣log⁡Y∣+(2Γ+12+log⁡(π))Y)+o(ρ2Y2),\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}e2D(ρ)=4πρ2Y(1−Y∣logY∣+(2Γ+21​+log(π))Y)+o(ρ2Y2),​ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as ρa2→0\rho a^2 \rightarrow 0ρa2→0 ρ a 2 → 0 . Here Y=∣log⁡(ρa2)∣−1Y= |\log (\rho a^2)|^{-1}Y=∣log(ρa2)∣−1 Y = | log ( ρ a 2 ) | - 1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee–Huang–Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.
We prove the following formula for the ground state energy density of a dilute Bose gas with density ρρρ in 222 dimensions in the thermodynamic limit \begin{align*} e^{\rm{2D}}(ρ) = 4πρ^2 Y\left(1 - Y \vert \log Y \vert + \left( 2Γ+ \frac{1}{2} + \log(π) \right) Y \right) + o(ρ^2 Y^{2}). \end{align*} Here Y=∣log⁡(ρa2)∣−1Y= |\log(ρa^2)|^{-1}Y=∣log(ρa2)∣−1 and aaa is the scattering length of the two-body potential. This result in 222 dimensions corresponds to the famous Lee-Huang-Yang formula in 333 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular it covers the crucial case of the hard core potential.
We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.