Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model

Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of -1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value -1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.