The free energy of the two-dimensional dilute Bose gas. I. Lower bound

TL;DR

This paper establishes a lower bound for the free energy of the two-dimensional dilute Bose gas, incorporating interaction effects and critical temperature, in the dilute and superfluid regime.

Contribution

It provides the first rigorous lower bound for the free energy of the 2D dilute Bose gas including interaction corrections near the superfluid transition.

Findings

Derived a lower bound for free energy in the dilute limit

Quantified the correction term involving scattering length and temperature

Connected free energy behavior to the Berezinskii--Kosterlitz--Thouless transition

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 $\rho$ and inverse temperature $\beta$ differs from the one of the non-interacting system by the correction term $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1 - \beta_{\mathrm{c}}/\beta]_+^2)$. Here $a$ is the scattering length of the interaction potential, $[\cdot]_+ = \max\{ 0, \cdot \}$ and $\beta_{\mathrm{c}}$ is the inverse Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. The result is valid in the dilute limit $a^2\rho \ll 1$ and if $\beta \rho \gtrsim 1$.