Magnetization plateaus in spin chains: ''Haldane gap'' for half-integer spins spins

We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, S, the magnetization curve can have plateaus and we argue that the magnetization per site m is topologically quantized as n(S-m)=integer at the plateaus, where n is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For S=3/2 and m=1/2, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a "Haldane gap phase" for half-integer S.