Transition from Bose-Einstein Condensate to Berezinskii-Kosterlitz-Thouless Phase

We obtain a phase diagram for a trapped two-dimensional ultra-cold Bose gas. We find a critical temperature above which the free energy of a state with a pair of vortices of opposite circulation is lower than the one for a vortex-free Bose-Einstein condensed ground state. We identify three distinct phases which are, in order of increasing temperature; a phase coherent Bose-Einstein condensate, a vortex pair plasma with a fluctuating condensate phase, and a thermal Bose gas. The existence of the vortex pair phase could be verified using current experimental setups.