Isotropic spin-1 chain with twisted boundary condition

The possible ground states and the phase transitions of a spin-1 chain with bilinear-biquadratic exchange are studied numerically. In order to see better the real asymptotic behavior of the infinite system and to determine the necessary degeneracies, a twisted boundary condition is used that reduces the finite-size effects. This calculation confirms that a second-order Ising-type phase transition occurs at the Takhtajan-Babujian point from the dimerized phase to the singlet Haldane phase. The transition at the Lai-Sutherland point, to a massless phase with three soft modes, is found to be of Kosterlitz-Thouless type.