We consider a variational approach to the Bose-Hubbard model based on
the Bogoliubov theory. We introduce the free energy functional for which
we prove the existence of minimizers. By analyzing their structure we
show the existence of a superfluid-Mott insulator phase transition at
positive temperatures. We also show that this model does not exhibit a
quantum phase transition. Joint work with Marcin NapiĆ³rkowski.