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

TL;DR

This paper investigates bipartite entanglement in heterogeneous spin-1 and spin-1/2 systems, highlighting the importance of higher-order correlations and providing general results applicable to various models with different symmetries.

Contribution

It introduces the role of triad and quad correlations in bipartite entanglement of heterogeneous spin systems, extending understanding beyond spin-1/2 models.

Findings

Higher-order correlations are crucial for entanglement in spin-1 systems.

Explicit formulas for reduced density matrices and negativity are derived.

Results apply to models with various symmetries and interaction types.

Abstract

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.