Reentrant Phenomenon in Quantum Phase Diagram of Optical Boson Lattice

We calculate the location of the quantum phase transitions of a bose gas trapped in an optical lattice as a function of effective scattering length $a_{\text{eff}}$ and temperature $T$. Knowledge of recent high-loop results on the shift of the critical temperature at weak couplings is used to locate a {\em nose} in the phase diagram above the free Bose-Einstein critical temperature $T_c^{(0)}$, thus predicting the existence of a reentrant transition {\em above} $T_c^{(0)}$, where a condensate should form when {\em increasing} $a_{\text{eff}}$. At zero temperature, the transition to the normal phase produces the experimentally observed Mott insulator.