Thermal phases of correlated lattice boson: a classical fluctuation theory

We present a method that generalises the standard mean field theory of correlated lattice bosons to include amplitude and phase fluctuations of the $U(1)$ field that induces onsite particle number mixing. This arises formally from an auxiliary field decomposition of the kinetic term in a Bose Hubbard model. We solve the resulting problem, initially, by using a classical approximation for the particle number mixing field and a Monte Carlo treatment of the resulting bosonic model. In two dimensions we obtain $T_c$ scales that dramatically improve on mean field theory and are within about 20% of full quantum Monte Carlo estimates. The `classical approximation' ground state, however, is still mean field, with an overestimate of the critical interaction, $U_c$, for the superfluid to Mott transition. By further including low order quantum fluctuations in the free energy functional we improve significantly on the $U_c$, and the overall thermal phase diagram. The classical approximation based method has a computational cost linear in system size. The methods readily generalise to multispecies bosons and the presence of traps.