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

TL;DR

This paper derives an explicit upper bound for the free energy density of a dilute two-dimensional Bose gas below the BKT transition, incorporating quasiparticle contributions with a specific dispersion relation.

Contribution

It provides a second order upper bound on the free energy of the 2D Bose gas using Bogoliubov theory, accounting for quasiparticle modes.

Findings

Derived an explicit upper bound for free energy density.

Captured quasiparticle contributions with a specific dispersion relation.

Applicable in the dilute regime below the BKT transition.

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 $\delta = 2 / (|\log(\rho a^2)| + \log |\log(\rho a^2)|)$.