Using recently developed algorithms for U(1) and SU(2) symmetric MPS with periodic boundary conditions (PBC), we investigate the phase diagram of the spin-1 bilinear-biquadratic Heisenberg (BBH) model with particular emphasis on the dimerized phase with and without quadratic Zeeman interaction.

Lowest-lying states are calculated for the whole range of the parameter θ for the BBH model of 10-100 sites without Zeeman effect. The results for the energies agree well with predictions of the Bethe ansatz and previous DMRG calculations (in particular, we obtain with high precision the Haldane gap for bilinear point θ = 0 and four lowest-lying states for the biquadratic point θ = -π/2). Furthermore, the results for the string correlator of a system of 100 sites and extrapolated results for the dimerization correlator also agree with earlier infinite-system calculations (the latter is calculated from two lowest-lying spin-0 states with different quasi-momenta, which form a degenerate doublet in the thermodynamic limit).

The question of the existence of the nematic phase close to SU(3) symmetric point θ = -3π/4 in the BBH model without Zeeman effect [1] is specifically addressed. To this end two extrapolated gaps are calculated: one between two lowest spin-0 states and the other between lowest spin-0 state and lowest spin-2 state. We confirm the absence of the nematic phase at least up to θ = -0.72π.

Furthermore, the bounds of the dimerized phase of the BBH model with quadratic Zeeman effect are determined, and the discrepancy of the results of different groups [2, 3] is judged. Our results indicate that the transition to the large-D phase occurs at least at D=0.3 at the biquadratic point.

 [1] K. Buchta, G. Fath, Ö. Legeza, J. Solyom, Phys. Rev. B 72, 054433 (2005).

 [2] K. Rodriguez, A. Argüelles, A. K. Kolezhuk, L. Santos, and T. Vekua, Phys. Rev. Lett. 106, 105302 (2011).

 [3] G. De Chiara, M. Lewenstein, A. Sanpera, Phys. Rev. B 84, 054451 (2011).