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