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

We discuss zero-temperature quantum spin chains in a uniform magnetic field. 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 $q (S - m)= \mbox{integer}$ at the plateaus, where $q$ is the period of the groundstate. 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 $S$.