Application of entanglement conditions to spin systems

There have been numerous studies of entanglement in spin systems. These have usually focused on examining the entanglement between individual spins or determining whether the state of the system is completely separable. Here, we present conditions that allow us to determine whether blocks of spins are entangled. We show that sometimes these conditions can detect entanglement better than conditions involving individual spins. We apply these conditions to study entanglement in spin-wave states, both when there are only a few magnons present and also at finite temperature.

Figure 1

The quantity $Q$ as a function of the interatomic distance, scaled by the lattice constant. A positive $Q$ indicates entanglement. Some large (positive) values of $Q$ are beyond the scope of the figure. The temperature is fixed at $\sqrt{2\mathit{JS}/(k_{\mathrm{B}}T)}=7$.

Figure 2

The quantity $Q$ as a function of the temperature for different interparticle distances $\Delta r/a=$ (a) 1 (solid line), (b) 3 (dashed line), (c) 10 (dotted line), and (d) 20 (dot-dashed line).

Figure 3

The quantity $Q$ as a function of the block size $m$, the number of spins in each block, for a fixed dimensionless temperature $\sqrt{2\mathit{JS}/(k_{\mathrm{B}}T)}=7$ and for $L=13$. The inset zooms in the part of the plot for small $m$.