Probing Evaporating Black Holes with Modular Flow in SYK

TL;DR

This paper investigates how modular flow affects fermionic correlation functions in the SYK model coupled to a bath, revealing non-trivial singularities and insights into entanglement and bulk-boundary correspondence in AdS2 black holes.

Contribution

It demonstrates the effects of modular flow on correlators in SYK models, connecting boundary singularities to bulk quantum extremal surfaces and entanglement wedge dynamics.

Findings

Modular flowed correlators exhibit non-trivial singularities at late times.

Correlations transfer between boundaries beyond a critical modular parameter.

Bulk modular flow can move fermions past the horizon, probing physics beyond the horizon.

Abstract

We study the effect of modular flow on correlation functions of fermions in the Sachdev-Ye-Kitaev (SYK) model coupled weakly to a bath, which we take to be another SYK model. The system and bath, together are prepared in the thermofield double (TFD) state, and we focus on the effect of modular flow generated by the reduced density matrix for the SYK system, obtained by tracing out the bath. We show, in the late time limit, that modular flowed correlators of two Majorana fermions, single-sided and two-sided, exhibit non-trivial singularities. Beyond a critical value of the modular parameter, the ``modular scrambling time", the singularity structure shows correlations being transferred from one boundary to the other. The calculations are performed by employing the replica trick in Euclidean time and appropriately analytically continuing to real time. Exploiting the connection between modular flow generators and SL$(2,{\mathbb R})$ boosts we use the microscopic picture to reconstruct the dual bulk modular flow in two-sided AdS$_2$ black hole spacetime. Fixed points of the flow allow to identify quantum extremal surfaces (QES) demarcating the entanglement wedge of the boundary system and the island. We show that bulk modular flow can move fermion insertions near the right boundary past the horizon leading to lightcone singularities in appropriately smeared boundary correlators, probing physics beyond the horizon.