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The large coordination number expansion of a lattice Bose gas at finite temperature

The expansion of the partition function for large coordination number $Z$ is a long standing method and has formerly been used to describe the Ising model at finite temperatures. We extend this approach and study the interacting Bose gas at finite temperatures. An analytical expression for the free energy is derived which is valid for weakly interacting and strongly interacting bosons. The transition line which separates the superfluid phase from Mott insulating/normal gas phase is shown for fillings $\langle\hat n\rangle=1$ and $\langle\hat n\rangle=2$. For unit filling, our findings agree qualitatively with Quantum Monte-Carlo results. Contrary to the well-known mean-field result, the shift of the critical temperature in the weakly interacting regime is apparent.