Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 2 / b / ii / Solution

There are two competing extremal-surface topologies. The disconnected candidate ${\gamma }_1\cup {\gamma }_2$ has area independent of the separation $L$ and gives $I_c=0$. A connected surface joining the two boundary circles can have smaller area when the gap is small; as $L\to 0$, short-distance entanglement across the nearby boundaries makes $I_c$ positive and divergent.

Increasing $L$ makes the connected candidate less favorable. At a critical separation $L=C(R_1,R_2)$ its area equals the disconnected area, and beyond that point the disconnected candidate is minimal. This entanglement-wedge phase transition gives