  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 Renyi entropies. Using Euclidean gravity, Fursaev computed the
entanglement Renyi 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 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.

High Energy Physics - Theory;Statistical Mechanics;Quantum Physics