We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence Nj tending to infinity in the whole range of possible p’s, in dimensions d≥1. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.