The Bogoliubov-Bose-Hubbard model: existence of minimizers and absence of quantum phase transition

We consider a variational approach to the Bose-Hubbard model based on Bogoliubov theory. We introduce the grand canonical and canonical free energy functionals for which we prove the existence of minimizers. By analyzing their structure we show the existence of a thermally driven phase transition by showing that the system is superfluid at sufficiently low temperatures and insulating at high temperatures. In particular, we show that this model does not exhibit a quantum phase transition.