Probing Evaporating Black Holes with Modular Flow in SYK

TL;DR

This work studies modular flow of Majorana fermions in two coupled SYK models prepared in a thermofield double state, with one system traced to form a bath. Using a replica trick in Euclidean time and a large-q limit, the authors derive both replica-diagonal and replica-non-diagonal saddles, unveiling modular-flow-induced singularities and a modular scrambling time that signals correlation transfer across boundaries. The SL$(2,\mathbb{R})$ structure of modular flow is matched to bulk AdS$_2$ dynamics, identifying fixed points as quantum extremal surfaces and constructing the island-entanglement wedge picture; bulk modular flow trajectories probe behind-horizon physics and yield lightcone singularities in smeared boundary correlators. The findings tie microscopic SYK calculations to bulk JT gravity, demonstrating maximal modular chaos and island formation, and provide a framework for probing evaporating-black-hole interiors via modular flow, with connections to BCFT analogies and future questions about infalling observers.

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.

Figures (15)

• Figure 1: Different trajectories under modular flow of the position of a right Majorana fermion inserted at a cutoff radial coordinate in AdS$_2$, at different temporal regimes $t_R$, in the sector with twist field $T_{R}$ in the right copy. The AdS$_{2}$ conformal boundaries are the dashed gray lines, and the yellow region represents the entanglement wedge of the right boundary points within a scrambling time. The location of the right QES, the fixed point of the flow, is labeled as $a_R$.
• Figure 2: Modular flowed trajectory (red line) of a Majorana fermion inserted on the right boundary of the cut off AdS$_2$ geometry at $t_R=0$ in the sector with twist field $T_{L}$ on the left copy. The positions of the AdS$_{2}$ conformal boundaries are denoted in dashed gray and the yellow region is the entanglement wedge of the left boundary points within a scrambling time. $a_L$ denotes the location of the left QES, which is the fixed point of the flow.
• Figure 3: On the left is the depiction of the state $\ket{{\rm TFD}(t)}$, obtained by evolution by $H_L$ in Euclidean time $\beta/2$ followed by evolution in real time $2t$. Dotted lines represent interactions between the $\chi$ and $\psi$ systems. The figure on the right represents the reduced density matrix $\rho$ obtained by tracing over the SYK$_\psi$ bath degrees of freedom.
• Figure 4: Depiction of the $n=3$ replica contour ${\cal C}$ with blue dashed lines showing cyclic identification of contours across different copies. Black dotted lines represent the interactions between the SYK$_\chi$ (blue) and SYK$_\psi$ (red) systems.
• Figure 5: The time contour $C_{*}$, divided into $C_1$ and $C_2$. One swap operator is inserted at $\tau = 0$ and another is inserted at $\tau = \beta/2+it$.