Entanglement Rényi entropies in holographic theories

Ryu and Takayanagi conjectured a formula for the entanglement (von Neumann) entropy of an arbitrary spatial region in an arbitrary holographic field theory. The von Neumann entropy is a special case of a more general class of entropies called Rényi entropies. Using Euclidean gravity, Fursaev computed the entanglement Rényi entropies (EREs) of an arbitrary spatial region in an arbitrary holographic field theory, and thereby derived the RT formula. We point out, however, that his EREs are incorrect, since his putative saddle points do not in fact solve the Einstein equation. We remedy this situation in the case of two-dimensional conformal field theories (CFTs), considering regions consisting of one or two intervals. For a single interval, the EREs are known for a general CFT; we reproduce them using gravity. For two intervals, the RT formula predicts a phase transition in the entanglement entropy as a function of their separation, and that the mutual information between the intervals vanishes for separations larger than the phase transition point. By computing EREs using gravity and CFT techniques, we find evidence supporting both predictions. We also find evidence that large $N$ symmetric product theories have the same EREs as holographic ones.

Figure 1

The two locally minimal surfaces for the boundary region $[u_1,v_1]\cup [u_2,v_2]$. The global minimum is the one on the left is when $x<1/2$, and the one on right when $x>1/2$, where $x$ is the cross-ratio defined in (4.5).

Figure 2

Modular parameter $\tau =il$ for the two-sheeted Riemann surface with branch points at 0, $x$, 1, $\infty$. The relation between $x$ and $l$ is given by Eq. (4.22).

Figure 3

Mutual Rényi information of intervals $[0,x]$ and $[1,\infty ]$ for a free scalar field of radius $R$, with (from bottom to top) $R^2=1$, 2, 4, 8, 16.

Figure 4

Coefficient of $c$ in the mutual Rényi information of intervals $[0,x]$ and $[1,\infty ]$ in a general holographic CFT. The two plots differ only in the scale of the vertical axis. In particular, the plot on the right shows that $I_1^{(2)}$ is nonzero (although quite small) for $x<1/2$.