A second order upper bound to the free energy of the two dimensional Bose gas

Abstract

We consider a two-dimensional Bose gas in the dilute regime where $\rho a^2$ is small. For temperatures below the Berezinskii-Kosterlitz-Thouless critical temperature, we derive an explicit upper bound for the free energy density using Bogoliubov theory. Our result captures the contribution of quasiparticle modes with dispersion relation $\sqrt{p^4 + 8\pi \rho\, \delta\, p^2}$ and where $\delta = 2 / (|\log(\rho a^2)| + \log |\log(\rho a^2)|)$.