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Here we investigate the phase diagram of the SO(n) bilinear-biquadraticquantum spin chain by studying the global quantum correlations of the groundstate. We consider the cases of n=3,4 and 5 and focus on the geometricentanglement in the thermodynamic limit. Apart from capturing all the knownphase transitions, our analysis shows a number of novel distinctive behaviorsin the phase diagrams which we conjecture to be general and valid for arbitraryn. In particular, we provide an intuitive argument in favor of an infiniteentanglement length in the system at a purely-biquadratic point. Our resultsare also compared to other methods, such as fidelity diagrams.

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