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.

Figure 1

Phase diagram of the SO($n$) bilinear biquadratic chain for (a) SO(3), (b) SO(4), and (c) SO(5). Black and gray dots indicate respectively known continuous and discontinuous transitions, whereas white dots indicate other interesting points of the system. The white dot close to $\theta =0$ corresponds to the MPS point $\tan \theta =1/n$ in all cases (AKLT for $n=3$).

Figure 2

GE of the SO($n$) bilinear-biquadratic model, for (a) SO(3), (b) SO(4), and (c) SO(5).

Figure 3

Different zooms to the GE diagram from Fig. 2a: (a) Around $\theta \approx 0.15\pi$ the GE is smooth (the line is a guide to the eye); (b) at the KT transition point $\theta =\pi /4$ the GE shows a cusp (the line is a guide to the eye); (c) the GE goes to zero at the first-order transition point $\theta =5\pi /4$, seemingly according to the continuous law $\mathcal{E}\sim \sqrt{(\theta -5\pi /4)}$ (continuous line); (d) at $\theta \approx 1.34\pi$ the GE shows a discontinuity, which eventually vanishes as the MPS bond dimension increases (black line is for $\chi =80$, blue line for $\chi =100$, red line for $\chi =120$, and the position of points has been omitted for the sake of clarity); (e) at the integrable point $\theta =3\pi /2$ the GE shows a cusp (the line is a guide to the eye); (f) around the second-order transition at $\theta =7\pi /4$ the GE shows a smooth change of behavior, compatible with $\mathcal{E}\sim a|{\theta }_c{-\theta |}^{\nu }$ with $\nu \sim 1$ (the line is a guide to the eye).

Figure 4

Different sections of the FD of the spin-1 bilinear-biquadratic spin chain, capturing a variety of different behaviors: (a) Region $\theta \in (5\pi /4,\pi /2)$; (b) region $\theta \in (0.3\pi ,0.7\pi )$; (c) region $\theta \in (1.15\pi ,1.35\pi )$.