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Communications in Mathematical Physics

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge, Leo Morin, Marco Olivieri

Published Feb 23, 2024• 10 citations
Publisher

Abstract

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.

DOI
10.1007/s00220-023-04907-2

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