Critical and noncritical long-range entanglement in Klein-Gordon fields

We investigate the entanglement between two spatially separated intervals in the vacuum state of a free one-dimensional Klein-Gordon field by means of explicit computations in the continuum limit of the linear harmonic chain. We demonstrate that the entanglement, which we quantify by the logarithmic negativity, is finite with no further need for renormalization. We find that in the critical regime, the quantum correlations are scale invariant as they depend only on the ratio of distance to length. They decay much faster than the classical correlations as in the critical limit long-range entanglement decays exponentially for separations larger than the size of the blocks, while classical correlations follow a power-law decay. With decreasing distance of the blocks, the entanglement diverges as a power law in the distance. The noncritical regime manifests richer behavior, as the entanglement depends both on the size of the blocks and on their separation. In correspondence with the von Neumann entropy also long-range entanglement distinguishes critical from noncritical systems.

Figure 1

(Color online) Critical HC: $\text{ln}{E}_{LN}$ as a function of $r\equiv d/l$. For $r>0.5$, the linear approximation practically coincides with the computed values ${\beta }_c\sim 2\sqrt{2}$. The dotted line is the overall estimation [Eq. (8)]. Upper inset: $\text{ln}\left(E_1/E_0\right)$ as a function of $\text{ln}r$. Lower inset: $\text{ln}{E}_{LN}$ as a function of $L$ for different values of $r$ [$D$ (the number of oscillators that separate the blocks) increases with $L$].

Figure 2

(Color online) Saturation in the noncritical chain. $\text{ln}{E}_{LN}$ as a function of $l$ for various values of $d_0$. The curves in the legend from top to bottom correspond to the curves shown from top to bottom in the figure. The points $E_{LN}\left(l=d_0\right)$ fit a linear curve.

Figure 3

(Color online) Noncritical chain. $\text{ln}{E}_{LN}$ as a function of $d$ for various values of $l_0$. The curves in the legend from top to bottom correspond to the curves shown from left to right in the figure. We observe the exponential decay with a quadratic function of $d$ (which also depends on $l_0$). In the intermediate saturation regime $l_0>d>1$, the decay is exponentially linear with $d$. Inset: power-law divergence in the $d\to 0$ limit (shown in log-log scale). The power is different than the $\frac{1}{3}$ in the critical limit and in general depends on $l_0$.

Figure 4

(Color online) Transition from critical to noncritical chain. $\text{ln}{E}_{LN}$ as a function of $d$ for constant values of $r=d/l$ ($l$ increases with $d$). The curves in the legend from top to bottom correspond to the curves shown from top to bottom in the figure. Note that all curves begin with the critical behavior as the entanglement is approximately constant. As $l$ becomes close to $m^{-1}=1$, the noncritical behavior emerges. All curves with $r<1$ coincide in correspondence with the saturation regime, which is independent of $l$. The curves with $d>l$ do not coincide and characterize the regime where $E_{LN}\sim \text{exp}\left[-{\beta }_{nc}\left(l\right)d^2\right]$.