Scaling of entanglement between separated blocks in spin chains at criticality

We compute the entanglement between separated blocks in certain spin models showing that at criticality this entanglement is a function of the ratio of the separation to the length of the blocks and can be written as a product of a power law and an exponential decay. It thereby interpolates between the entanglement of individual spins and blocks of spins. It captures features of correlation functions at criticality as well as the monogamous nature of entanglement. We exemplify invariant features of this entanglement to microscopic changes within the same universality class. We find this entanglement to be invariant with respect to simultaneous scale transformations of the separation and the length of the blocks. As a corollary, this Rapid Communication estimates the entanglement between separated regions of those quantum fields to which the considered spin models map at criticality.

Figure 1

(Color online) Schematic of different types of entanglement at a quantum critical point. Panel (a): critical behavior of entanglement of pairs of spins, exhibiting very limited range. Panel (b): entanglement entropy of two contiguous and complementary blocks of spins, obeying a universal scaling law at criticality. Panel (c): entanglement between distant blocks of spins as a function of their size $\Delta$ and separation $x$ in a spin chain of finite but large length $N=2\Delta +x$.

Figure 2

Scaling of distant block entanglement with system size at and in the vicinity of the critical point $\lambda -{\lambda }_{\text{c}}=0$ of the transverse Ising model [$\gamma =1$ in Eq. (1)]. The data correspond to a fixed ratio $\mu =\frac{2}{3}$ and different system sizes $N=32$ (open circles), $N=64$ (squares), $N=128$ (diamonds), and $N=256$ (filled circles). The crossing of dashed lines highlights the converged, scale invariant point for $N\gtrsim 128$, which is $\left(\lambda ,\mathcal{N}\right)\cong \left({\lambda }_c,0.052\right)$ for this ratio.

Figure 3

Negativity $\mathcal{N}$ as a function of the ratio $\mu =\frac{x}{\Delta }$, for the critical $XX$ at vanishing field (diamonds), critical $XY$ (open circles), and critical Ising (filled circles) models. The data for the critical $XX$ model are fitted according to our ansatz, revealing the parameters $h=0.47$ and $\alpha =0.96$ (dashed line). The critical Ising model is fitted accordingly with parameters $h=0.38$ and $\alpha =1.68$. All data correspond to selected subsets of possible ratios $\mu$, in favor of visibility.