Ordering and Criticality in Spin-1 Chains

The $T=0$ behavior of the antiferromagnetic, nearest-neighbor, spin-1 chain is studied in a space of interactions which includes exchange anisotropy, bond alternation, and biquadratic exchange. We find that bond alternation can stabilize N\'eel order; that with sufficient biquadratic coupling the ground state may be spontaneously dimerized and simultaneously critical; and that there is a line of multicritical points where the N\'eel-ordered phase touches the manifold of isotropic coupling. Predictions of Affleck and Haldane regarding the criticality and ordering in these models are tested.