Holographic Entropy — Ryu-Takayanagi

Boundary Interval

Interval center (angle) 0°

Interval half-length 60°

Second interval (for phase transition) 180°

Show two intervals

Entropy Measurements

S(A) (connected)—

S(A) (disconnected)—

S(A∪B) (RT minimum)—

Geodesic length—

Phase—

Entropy vs Interval Size

S(A) (connected)

Complement

The Ryu-Takayanagi formula (2006) equates the entanglement entropy of a boundary region A to the area of the minimal bulk geodesic γ_A anchored at ∂A. In AdS₂/CFT₁, geodesics are semicircles in the Poincaré disk. When two intervals A, B are large enough to overlap, the minimal surface "jumps" from connected to disconnected configuration — a geometric phase transition encoding quantum information scrambling.