Phase diagram of the SO(n) bilinear-biquadratic chain from many-body entanglement

Here we investigate the phase diagram of the SO(n) bilinear-biquadratic quantum spin chain by studying the global quantum correlations of the ground state. We consider the cases of n=3,4, and 5 and focus on the geometric entanglement in the thermodynamic limit. Apart from capturing all the known phase transitions for these cases, our analysis shows a number of distinctive behaviors in the phase diagrams which we conjecture to be general and valid for arbitrary n. In particular, we provide an intuitive argument in favor of an infinite entanglement length in the system at a purely biquadratic point. Our results are also compared to those of other methods, such as fidelity diagrams.