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 $n(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}$ 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 $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2$ and $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, 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 $S$.