The first reliable analytic calculation of the phase diagram of the Bose gas on a d -dimensional lattice with on-site repulsion is presented. In one dimension, the analytic calculation is in excellent agreement with the numerical Monte Carlo results. In higher dimensions, the deviations from the Monte Carlo calculations are larger, but the correct shape of the Mott-insulator lobes is still obtained. Explicit expressions for the energy of the Mott and the "defect" phase are given in a strong-coupling expansion.
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