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arXiv:1901.00539 · Communications in Mathematical Physics

A Simple 2nd Order Lower Bound to the Energy of Dilute Bose Gases

Birger Brietzke, Søren Fournais, Jan Philip Solovej

Published Mar 12, 2020• 18 citations
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Abstract

For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound $$e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)$$ .

DOI
10.1007/s00220-020-03715-2

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For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1−Cρa3 )e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)e(ρ)≥4πaρ2(1−Cρa3​) .
For a dilute system of non-relativistic bosons interacting through a positive L1L^1L1 potential vvv with scattering length aaa we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1+12815πρa3+o(ρa3 ))e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))e(ρ)≥4πaρ2(1+15π​128​ρa3​+o(ρa3​)), 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, L1L^1L1-potential vvv with scattering length aaa we prove that the ground state energy density satisfies the bound e(ρ)≥4πaρ2(1+12815πρa3+o(ρa3))e(\rho) \ge 4\pi a \rho^2(1 + \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} + o(\sqrt{\rho a^3}))e(ρ)≥4πaρ2(1+15π​128​ρa3​+o(ρa3​)), thereby proving the Lee-Huang-Yang formula for the energy density.
Consider NNN bosons in a finite box $\ensuremath{\Lambda}={[0,L]}^{3}\ensuremath{\subset}{\mathbb{R}}^{3}$ interacting via a two-body non-negative soft potential $V=\ensuremath{\lambda}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ fixed and $\ensuremath{\lambda}>0$ small. We will take the limit $L,N\ensuremath{\rightarrow}\ensuremath{\infty}$ by keeping the density $\ensuremath{\varrho}=N∕{L}^{3}$ fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle $\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\leqslant}4\ensuremath{\pi}\ensuremath{\varrho}a[1+(128∕15\sqrt{\ensuremath{\pi}}){(\ensuremath{\varrho}{a}^{3})}^{1∕2}{S}_{\ensuremath{\lambda}}]+O({\ensuremath{\varrho}}^{2}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho}\ensuremath{\mid})$, as $\ensuremath{\varrho}\ensuremath{\rightarrow}0$, with a constant satisfying $1\ensuremath{\leqslant}{S}_{\ensuremath{\lambda}}\ensuremath{\leqslant}1+C\ensuremath{\lambda}$. Here aaa is the scattering length of VVV and thus depends on $\ensuremath{\lambda}$. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that ${S}_{\ensuremath{\lambda}}=1$ independent of $\ensuremath{\lambda}$.
We consider a gas of bosons interacting through a hard-sphere potential with radius a\frak{a}a in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term 4πρa4πρ\frak{a}4πρa and shows that corrections are smaller than Cρa(ρa3)1/2C ρ\frak{a} (ρ\frak{a}^3)^{1/2}Cρa(ρa3)1/2, for a sufficiently large constant C>0C > 0C>0. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order ρa(ρa3)1/2ρ\frak{a} (ρ\frak{a}^3)^{1/2}ρa(ρa3)1/2, in agreement with the Lee-Huang-Yang prediction.