Interacting Bose Gas in an Optical Lattice

A grand canonical system of hard core bosons in an optical lattice is considered. The bosons can occupy randomly N equivalent states at each lattice site. The limit N (infinity) is solved exactly in terms of a saddle point integration, representing a weakly interacting Bose gas. In the limit N (infinty) there is only a condensate if the fugacity of the Bose gas is larger than 1. Corrections in 1/N increase the total density of bosons but suppress the condensate. This indicates a depletion of the condensate due to increasing interaction at finite values of N.