Do bosons condense in a homogeneous magnetic field?

It has been known since the rigorous results by Angelescu, Corciovei, and Schafroth, that there is no Bose-Einstein Condensation (BEC) of the three-dimensional Perfect Bose Gas (PBG) in a homogeneous magnetic field. The main result of our work is that the answer can become positive if the bosons are simultaneously embedded in a periodic external potential. We show that it is true for PBG, as well as for the Bose gas with a mean-field repulsive particle interaction. In cooperation with Philippe Briet, Université de Toulon et du Var and with Valentin Zagrebnov, Université de la Méditerranées