Free-kick condition for entanglement entropy in higher curvature gravity

In order to compute the entanglement entropy for a given region in a theory with an Einstein gravity dual, the Ryu-Takayanagi prescription tells us that we must compute the area of an extremal surface anchored to the entangling region. However, if the dual gravity theory receives higher-curvature corrections we are compelled to extremize a quantity which is no longer given by the area but a higher-derivative functional. Hence, in order to find the extremal surface that yields the correct value of the entanglement entropy, we must include an additional boundary condition to the problem. We claim that the additional condition can be fixed by demanding that the relationship between the bulk depth and the size of the entangling region is the one induced by geodesics, we call this the free-kick condition. We implement this prescription in the computation of the entanglement entropy of the hairy black hole in new massive gravity, and find a perfect agreement with conformal field theory expectations.

Figure 1

The Ryu-Takayanagi prescription; $A$ is the entangling region, and $\mathrm{\Sigma }$ is the extremal surface in the asymptotically AdS background.

Figure 2

The comparison between geodesics in the BTZ (dashed) and OTT (solid purple) geometries; the radial coordinate is compactified using the map $r\to \arctan (r)$. The outer (solid black) circle represents asymptotic infinity, and the inner (dashed) circle the horizon radius.

Figure 3

Conformal diagram of entangling curves for different depths in the BTZ geometry.

Figure 4

Conformal diagram of extremal curves in the OTT geometry for different concavities (solid purple), geodesic curve shadow (gray), and the free-kick extremal curve (dashed).