2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/21110A 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 $N\to\infty$.12 pages, 4 figures, talk given at LPHYS '02Statistical Mechanicstext