Ground state energy of the two-dimensional weakly interacting Bose gas: First correction beyond Bogoliubov theory

We consider the grand potential $\mathrm{\Omega }$ of a two-dimensional weakly interacting homogeneous Bose gas at zero temperature. Building on a number-conserving Bogoliubov method for a lattice model in the grand canonical ensemble, we calculate the next order term as compared to the Bogoliubov prediction, in a systematic expansion of $\mathrm{\Omega }$ in powers of the parameter measuring the weakness of the interaction. Our prediction is in very good agreement with recent Monte Carlo calculations.