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arXiv:1305.0657 · Journal of the Physical Society of Japan

Ground-State Energy and Condensate Density of a Dilute Bose Gas Revisited

Kazumasa Tsutsui, Takafumi Kita

Published May 20, 2013• 3 citations

Abstract

The ground-state energy per particle $E/N$ and condensate density $n_0$ 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 $c_{ip}\sim O(1)$ to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as $E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]$ and $n_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}$, where $a$, $n$, and $m$ are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for $c_{ip}$; the third-order perturbation expansion yields $c_{ip}=0.412$.

DOI: 10.7566/jpsj.82.063001

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The ground-state energy per particle

E/N

and condensate density

n_0

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

c_{ip}\sim O(1)

to the expressions reported by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957) ] as

E/N=(2π\hbar^2 an/m)[1+(128/15\sqrt{π}+16c_{ip}/5)\sqrt{a^3n}]

and

n_0/n=1-(8/3\sqrt{π}+c_{ip})\sqrt{a^3n}

, where

a

,

n

, and

m

are are the s-wave scattering length, particle density, and particle mass, respectively. We present a couple of estimates for

c_{ip}

; the third-order perturbation expansion yields

c_{ip}=0.412

.

For a dilute system of non-relativistic bosons interacting through a positive

L^1

potential

v

with scattering length

a

we prove that the ground state energy density satisfies the bound

e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))

, thereby proving the Lee-Huang-Yang formula for the energy density.

For a dilute system of non-relativistic bosons interacting through a positive, compactly supported,

L^1

-potential

v

with scattering length

a

we prove that the ground state energy density satisfies the bound

e(\rho) \ge 4\pi a \rho^2(1 + \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} + o(\sqrt{\rho a^3}))

, thereby proving the Lee-Huang-Yang formula for the energy density.

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.

We prove the following formula for the ground state energy density of a dilute Bose gas with density

ρ

in

2

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(ρa^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.

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.

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density

\rho

ρ in 2 dimensions in the thermodynamic limit

\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}

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

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-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.