Entanglement in Spin Chains and Lattices with Long-Range Ising-Type Interactions

We consider $N$ initially disentangled spins, embedded in a ring or $d$-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit $N\to \infty$. We provide a sufficient condition when bipartite entanglement between blocks of $L$ neighboring spins and the remaining system saturates and determine $S_L$ analytically for special configurations. We find an unbounded increase of $S_L$ as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large $N$, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.

Figure 1

Spin chain with $N={10}^5$, $t=0.3\pi$, and $f(r_{kl})=r^{-\alpha }$ for different $\alpha$. (a) Maximal two-point correlation $Q_{\max }^{i,j}$ as a function of the distance $\Vert i-j\Vert$ for $\alpha =1/3$ (dashed curve) and $\alpha =3$ (solid curve). (b) Exact values [upper bounds] of entropy of entanglement $S_L$ as a function of block size $L$ for $\alpha =1/3$ (▵ [○]) and $\alpha =3$ (◊ [□]).

Figure 2

Dynamics of entanglement for chain with $N={10}^5$ and $f(r_{kl})=r_{kl}^{-3}$. (a) Entropy of entanglement $S_L$ for blocks $L=1$ (bottom) up to $L=7$ (top). (b) Two-point correlations $Q_{\max }^{i,j}$ for $\Vert i-j\Vert =5$ (solid line) and $E_{\mathrm{M}\mathrm{W}}$ (dashed line).