Superfluid to normal fluid phase transition in the Bose gas trapped in two dimensional optical lattices at finite temperature

We develop the Hartree-Fock-Bogoliubov theory at finite temperature for Bose gas trapped in the two dimensional optical lattices. The on-site energy is considered low enough that the gas presents superfluid properties. We obtain the condensate density as function of the temperature neglecting the anomalous density in the thermodynamics equations. The condensate fraction provide two critical temperature. Below the temperature $T_{C1}$ there is one condensate fraction. Above two possible fractions merger up to the critical temperature $T_{C2}$. Then the gas provides an first order transition at temperature above $T_{C2}$ where the condensate fraction is null. We resume by a finite-temperature phase diagram where can be identify three domains: the normal fluid, the superfluid and the superfluid with two possible condensate fractions.