Entangled spin clusters: Some special features

In this paper, we study three specific aspects of entanglement in small spin clusters. We first study the effect of inhomogeneous exchange coupling strengths on the entanglement properties of the $S=\frac{1}{2}$ antiferromagnetic linear chain tetramer compound $\mathrm{Na}\mathrm{Cu}\mathrm{As}{\mathrm{O}}_4$. The entanglement gap temperature $T_E$ is found to have a nonmonotonic dependence on the value of $\alpha$, the exchange coupling inhomogeneity parameter. We next determine the variation of $T_E$ as a function of $S$ for a spin dimer, a trimer, and a tetrahedron. The temperature $T_E$ is found to increase as a function of $S$ but the scaled entanglement gap temperature $t_E$ goes to zero as $S$ becomes large. Last, we study a spin-1 dimer compound to illustrate the quantum complementarity relation. We show that in the experimentally realizable parameter region, magnetization and entanglement plateaus appear simultaneously at low temperatures as a function of the magnetic field. Also, the sharp increase in one quantity as a function of the magnetic field is accompanied by a sharp decrease in the other so that the quantum complementarity relation is not violated.

Figure 1

Concurrence $C_{12}$ as a function of temperature for $\alpha =0.4$ and $\frac{J}{k_b}=92.7\mathrm{K}$.

Figure 2

Plots of $T_c^{12}$ and $T_c^{23}$ vs $\alpha$. $T_c^{kl}$ is the critical entanglement temperature for the pair of spins at sites $k$ and $l$.

Figure 3

Plot of $T_E$ (curve a) and $T_c^{\chi }$ (curve b) for the linear chain tetramer as a function of $\alpha (\frac{J}{k_B}=92.7\mathrm{K})$.

Figure 4

Variation of the entanglement gap temperature $T_E$ with $S$ for dimers (star), trimers (solid triangle), and tetrahedra (solid square). The inset shows the variation of the scaled entanglement gap temperature $t_E$ with $S$.

Figure 5

Plots of $P$, $Q$, and $P+Q$ as a function of $\frac{B}{J}$ in the case of a spin-1 dimer ($\delta =1$, $d=0$, and $\beta J=3$).

Figure 6

Plots of $P$, $Q$, and $P+Q$ as a function of $\frac{B}{J}$ in the case of a spin-1 dimer ($\delta =1$, $d=0$, and $\beta J=20$).