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We present a detailed derivation of a multi-site mean-field theory (MSMFT)used to describe the Mott-insulator to superfluid transition of bosonic atomsin optical lattices. The approach is based on partitioning the lattice intosmall clusters which are decoupled by means of a mean field approximation. Thisapproximation invokes local superfluid order parameters defined for each of theboundary sites of the cluster. The resulting MSMFT grand potential has anon-trivial topology as a function of the various order parameters. Anunderstanding of this topology provides two different criteria for thedetermination of the Mott insulator superfluid phase boundaries. We apply thisformalism to $d$-dimensional hypercubic lattices in one, two and threedimensions, and demonstrate the improvement in the estimation of the phaseboundaries when MSMFT is utilized for increasingly larger clusters, with thebest quantitative agreement found for $d=3$. The MSMFT is then used to examinea linear dimer chain in which the on-site energies within the dimer have anenergy separation of $\Delta$. This system has a complicated phase diagramwithin the parameter space of the model, with many distinct Mott phasesseparated by superfluid regions.

Multi-site mean-field theory for cold bosonic atoms in optical lattices