Magnetization Plateaus in Spin Chains: “Haldane Gap” for Half-Integer 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) 0ex{0ex}=0ex{0ex}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 S0ex{0ex}=0ex{0ex}3/2 and m0ex{0ex}=0ex{0ex}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.