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Thermodynamics of the Bose-Hubbard model in a Bogoliubov+U theory

We derive the Bogoliubov+U formalism to study the thermodynamical properties of the Bose-Hubbard model. The framework can be viewed as the zero-frequency limit of bosonic dynamical mean-field theory (B-DMFT), but equally well as an extension of the mean-field decoupling approximation in which pair creation and annihilation of depleted particles is taken into account. The self-energy on the impurity site is treated variationally, minimizing the grand potential. The theory containing just three parameters that are determined self-consistently reproduces the T=0 phase diagrams of the three-dimensional and two-dimensional Bose-Hubbard model with an accuracy of 1% or better. The superfluid to normal transition at finite temperature is also reproduced well and only slightly less accurately than in B-DMFT.