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arXiv · 2206.11100

The Ground State Energy of a Two-Dimensional Bose Gas

S. Fournais·T. Girardot·L. Junge·L. Morin·M. Olivieri
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Abstract

We prove the following formula for the ground state energy density of a dilute Bose gas with density $\rho$ in $2$ dimensions in the thermodynamic limit \begin{align*} e^{\rm{2D}}(\rho) = 4\pi \rho^2 Y\left(1 - Y \vert \log Y \vert + \left( 2\Gamma + \frac{1}{2} + \log(\pi) \right) Y \right) + o(\rho^2 Y^{2}). \end{align*} Here $Y= |\log(\rho a^2)|^{-1}$ 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.

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S. Fournais, T. Girardot, L. Junge, L. Morin, M. Olivieri. 2022-06-22. The Ground State Energy of a Two-Dimensional Bose Gas. https://arxiv.org/abs/2206.11100

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