The free energy of the two-dimensional dilute Bose gas. II. Upper bound

Abstract

We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for $a^2\rho \ll 1$ and $\beta \rho \gtrsim 1$ the free energy per unit volume differs from the one of the non-interacting system by at most $4\pi {\rho }^2|\ln a^2\rho {|}^{-1}(2-[1-{\beta }_{\mathrm{c}}/\beta {]}_{+}^2)$ to leading order, where $a$ is the scattering length of the two-body interaction potential, $\rho$ is the density, $\beta$ the inverse temperature and ${\beta }_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved in \cite{DMS19} this shows equality in the asymptotic expansion.

LaTeX, 24 pages; final version, to appear in J. Math. Phys