THE FREE ENERGY OF THE TWO-DIMENSIONAL DILUTE BOSE GAS. I. LOWER BOUND

Abstract

We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density$\unicode[STIX]{x1D70C}$and inverse temperature$\unicode[STIX]{x1D6FD}$differs from the one of the noninteracting system by the correction term$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2-[1-\unicode[STIX]{x1D6FD}_{\text{c}}/\unicode[STIX]{x1D6FD}]_{+}^{2})$. Here,$a$is the scattering length of the interaction potential,$[\cdot ]_{+}=\max \{0,\cdot \}$and$\unicode[STIX]{x1D6FD}_{\text{c}}$is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit$a^{2}\unicode[STIX]{x1D70C}\ll 1$and if$\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70C}\gtrsim 1$.