An explicit upper bound to the free energy density of the dilute 2D Bose gas below the BKT transition is obtained via Bogoliubov theory with quasiparticles obeying dispersion sqrt(p^4 + 8 pi rho delta p^2) where delta involves double logarithms of rho a^2.
P.Solovej, Mathematical physics of dilute Bose gases,Comptes Rendus
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A second order upper bound to the free energy of the two dimensional Bose gas
An explicit upper bound to the free energy density of the dilute 2D Bose gas below the BKT transition is obtained via Bogoliubov theory with quasiparticles obeying dispersion sqrt(p^4 + 8 pi rho delta p^2) where delta involves double logarithms of rho a^2.