The Ground State Energy of a Two-Dimensional Bose Gas

We prove the following formula for the ground state energy density of a dilute Bose gas with density \(\rho \) $\rho$ in 2 dimensions in the thermodynamic limit \(\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}\) $$\begin{array}{c}\hfill e^{\text{2D}}(\rho )=4\pi {\rho }^{2}Y(1-Y|logY|+(2\mathrm{\Gamma }+\frac{1}{2}+log(\pi ))Y)+o({\rho }^2{Y}^2),\end{array}$$ as \(\rho a^2 \rightarrow 0\) $\rho a^2\to 0$ . Here \(Y= |\log (\rho a^2)|^{-1}\) $Y=|log(\rho a^2){|}^{-1}$ 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.