  The Heisenberg model for spin-1 bosons in one dimension presents many
different quantum phases including the famous topological Haldane phase. Here
we study the robustness of such phases in front of a SU(2) symmetry breaking
field as well as the emergence of novel phases. Previous studies have analyzed
the effect of such uniaxial anysotropy in some restricted relevant points of
the phase diagram. Here we extend those studies and present the complete phase
diagram of the spin-1 chain with uniaxial anysotropy. To this aim, we employ
the density matrix renormalization group (DMRG) together with analytical
approaches. The complete phase diagram can be realized using ultracold spinor
gases in the Mott insulator regime under a quadratic Zeeman effect.
