Quantum correlations and coherence in spin-1 Heisenberg chains

We explore quantum and classical correlations along with coherence in the ground states of spin-1 Heisenberg chains, namely the one-dimensional XXZ model and the one-dimensional bilinear biquadratic model, with the techniques of density matrix renormalization group theory. Exploiting the tools of quantum information theory, that is, by studying quantum discord, quantum mutual information, and three recently introduced coherence measures in the reduced density matrix of two nearest neighbor spins in the bulk, we investigate the quantum phase transitions and special symmetry points in these models. We point out the relative strengths and weaknesses of correlation and coherence measures as figures of merit to witness the quantum phase transitions and symmetry points in the considered spin-1 Heisenberg chains. In particular, we demonstrate that, as none of the studied measures can detect the infinite-order Kosterlitz-Thouless transition in the XXZ model, they appear to be able to signal the existence of the same type of transition in the biliear biquadratic model. However, we argue that what is actually detected by the measures here is the SU(3) symmetry point of the model rather than the infinite-order quantum phase transition. Moreover, we show in the XXZ model that examining even single site coherence can be sufficient to spotlight the second-order phase transition and the SU(2) symmetry point.

Figure 1

Mutual information versus the anisotropy parameter $\mathrm{\Delta }$ in the one-dimensional spin-1 XXZ model. Different phases, transition points, and the $\text{SU}\left(2\right)$ symmetry point are shown.

Figure 2

Quantum discord versus the anisotropy parameter $\mathrm{\Delta }$ in the one-dimensional spin-1 XXZ model. Different phases, transition points, and the $\text{SU}\left(2\right)$ symmetry point are shown.

Figure 3

Quantum coherence measures as quantified by $l_1$ norm of coherence, relative entropy of coherence, and $K$ coherence versus the anisotropy parameter $\mathrm{\Delta }$ in the one-dimensional spin-1 XXZ model. Different phases, transition points, and the $\text{SU}\left(2\right)$ symmetry point are shown.

Figure 4

Mutual information versus the parameter $\theta$ in the one-dimensional spin-1 bilinear biquadratic model. Different phases, transition points, and the $\text{SU}\left(3\right)$ symmetry point are shown.

Figure 5

Quantum discord versus the parameter $\theta$ in the one-dimensional spin-1 bilinear biquadratic model. Different phases, transition points, and the $\text{SU}\left(3\right)$ symmetry point are shown.