Entanglement islands and cutoff branes from path-integral optimization

TL;DR

This paper explores how Weyl transformations optimize path integral computations in holographic CFT$_2$, revealing connections to entanglement islands, cutoff branes, and the AdS/BCFT correspondence, with results matching holographic entanglement measures.

Contribution

It demonstrates that path-integral optimization leads to cutoff branes acting as entanglement islands, providing a geometric understanding of negative mutual information in holographic BCFTs.

Findings

The cutoff brane forms a circle passing through interval endpoints.

Entanglement entropy matches RT formula with cutoff brane anchoring.

Negative mutual information arises naturally in the island phase.

Abstract

Recently it was proposed that, the AdS/BCFT correspondence can be simulated by a holographic Weyl transformed CFT$_2$, where the cut-off brane plays the role of the Karch-Randall (KR) brane \cite{Basu:2022crn}. In this paper, we focus on the Weyl transformation that optimizes the path integral computation of the reduced density matrix for a single interval in a holographic CFT$_2$. When we take the limit that one of the endpoint of the interval goes to infinity (a half line), such a holographic Weyl transformed CFT$_2$ matches the AdS/BCFT configuration for a BCFT with one boundary. Without taking the limit, the induced cutoff brane becomes a circle passing through the two endpoints of the interval. We assume that the cutoff brane also plays the same role as the KR brane in AdS/BCFT, hence the path-integral-optimized purification for the interval is in the island phase. This explains the appearance of negative mutual information observed in \cite{Camargo:2022mme}. We check that, the entanglement entropy and the balanced partial entanglement entropy (BPE) calculated via the island formulas, exactly match with the RT formula and the entanglement wedge cross-section (EWCS), which are allowed to anchor on the cutoff brane.