Entanglement Entropy for Probe Branes

TL;DR

The paper addresses the challenge of computing entanglement entropy in holographic theories with probe branes by deriving a leading-order backreaction formula that reduces to a compact double integral S_A = (π T_0) ∫ d^{d-1} w √γ ∫ d^{n+1} z √g_I T^{μν}_{min} G_{μνρσ} T^{ρσ}_{probe}. This approach is validated against two solvable toy models where the full backreaction is known, and subsequently mapped to top-down D3/D7 and D3/D5 systems to obtain EE for sphere and strip entangling regions, with the spherical result agreeing with the Casini-Huerta-Myers/Casini-O'Bannon method. The work shows that, at leading order, the EE correction depends mainly on the probe and minimal-surface stress tensors via the gravitational Green's function, and that the internal space often does not affect the EE in these setups. The derived framework provides a practical tool for exploring EE in a wide range of probe-brane holographic theories and can be extended to more general entangling surfaces and higher-derivative gravity corrections.

Abstract

We give a prescription for calculating the entanglement entropy in holographic probe brane systems by systematically taking the leading order backreaction of the probe brane into account. We find a simple compact double integral formula, which is insensitive to many details of the backreaction, most notably the internal space or the non-metric fields sourced by the probe. We validate our method by comparing to exact results in solvable toy models. We also determine the entanglement entropies for a sphere and a strip in the top-down D3/D7 and D3/D5 system. For the sphere the entanglement entropy has also been obtained by other methods and we find perfect agreement.