  A grand-canonical system of interacting bosons is considered to study phase
transitions of ultracold atoms in an optical lattice. The phase diagram is
discussed in terms of a matrix-like order parameter, representing a symmetric
phase (Mott insulator) and a symmetry-broken phase (Bose-Einstein condensate).
An exact solution was found by introducing $N$ colors and taking the limit
$N\to\infty$.
