Entanglement in heterogeneous spin-$(1, \frac 12)$ and homogeneous spin-1 systems

We study the bipartite entanglement of two general classes of heterogeneous spin-($1,\frac 12$) and homogeneous spin-1 systems. By employing the spin correlation functions, we obtain the reduced two-spin density matrix (DM) and the negativity for these two classes of quantum spin models. We show explicitly that in addition to the one and two-point correlations, the triad and quad correlations ($t_{\alpha\beta}^{\delta}=\la S_{\alpha}S_{\beta}s_{\delta}\ra$ and $q_{\alpha\beta}^{\delta\gamma}=\la S_\alpha S_\beta s_\delta s_\gamma\ra$ where $\alpha, \beta, \delta, \gamma=\pm, z$) play crucial role in the bipartite entanglement between spins $s>1/2$. These correlations represent the spin $\frac 12$-quadrupole and quadrupole-quadrupole correlations, respectively. These correlations do not appear in the spin-$\frac 12$ models. Our results are general and applicable to the different several models of interest with higher reflectional, translational, spin-flip and U(1) symmetries. The entanglement of many attractive models are investigated.