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)0ex0ex=0ex0exinteger 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 S0ex0ex=0ex0ex3/2 and m0ex0ex=0ex0ex1/2, 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.