Extended massless phase and the Haldane phase in a spin-1 isotropic antiferromagnetic chain

We study the phase diagram of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland (ULS) model. This is done with the help of a level-one SU (3) Wess-Zumino-Witten model with certain marginal perturbations. We find that the renormalization-group flow has infrared stable and unstable trajectories divided by a critical line on which the ULS model is located. The infrared unstable trajectory produced by a marginally relevant perturbation generates an exponential mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at the ULS point is identified with the Berezinski -Kosterlitz-Thouless type. Our results support recent numerical studies by F\'ath and S\'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU () -symmetric and asymmetric models in the massless phase.