Entanglement entropy of spherical domains in anti–de Sitter space

It was proposed by Ryu and Takayanagi that the entanglement entropy in conformal field theory (CFT) is related through the AdS/CFT correspondence to the area of a minimal surface in the bulk. We apply this holographic geometrical method of calculating the entanglement entropy to study the vacuum case of a CFT which is holographically dual to empty anti-de Sitter (AdS) spacetime. We present all possible minimal surfaces spanned on one or two spherical boundaries at AdS infinity. We give exact analytical expressions for the regularized areas of these surfaces and identify finite renormalized quantities. In the case of two disjoint boundaries the existence of two different phases of the entanglement entropy is confirmed. A trivial phase corresponds to two disconnected minimal surfaces, while the other one corresponds to a tube connecting the spherical boundaries. A transition between these phases is reminiscent of the finite temperature deconfinement transition in the CFT on the boundary. The exact analytical results are thus consistent with previous numerical and approximate computations. We also briefly discuss the character of a spacetime extension of the minimal surface spanned on two uniformly accelerated boundaries.

Figure 1

Hyperplane corresponding to the circular boundary at infinity. The hyperplane (the hypersurface of zero extrinsic curvature) reaches infinity at the circular boundary which divides infinity into two domains. The Poincaré spherical (left) and spherical half-space (right) representation of hyperbolic space is shown.

Figure 2

Minimal surfaces spanned on circular boundaries. The surfaces are visualized in Poincaré spherical (top) and Poincaré half-space (bottom) models. (a) The tubelike surface spanned on two disjoint boundaries. (b) The surface spanned on two semicircles joining opposite poles. (c) The surfaces spanned on two touching circles.

Figure 3

Renormalized area of the minimal surface spanned on two boundaries. (a) Relation between the distance $s$ of the boundaries and the closest approach $P_0$ of the tube to the axis. (b) Renormalized area of the tube as a function of $P_0$. (c) Renormalized area as a function of the distance $s$.

Figure 4

World sheets of minimal surfaces. The vertical cylinder represents a $3+1$ dimensional AdS spacetime with the angular direction $\phi$ suppressed. The world sheet of one circular boundary is thus reduced only to two curves at infinity. One circular boundary is localized in the left static region, the other in the right one. (a) Two uniformly accelerated hyperplanes spanned on these circular boundaries. Killing horizons are indicated. (b) The world sheet of the tubelike minimal surface joining the same circular boundaries.