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arXiv:2406.05176 · Journal of High Energy Physics

Quantum extremal modular curvature: modular transport with islands

Lars Aalsma, Cynthia A Keeler, Claire Zukowski

Published Oct 1, 2024• 2 citations
arXivHTMLPDF

Abstract

A bstract Modular Berry transport is a useful way to understand how geometric bulk information is encoded in the boundary CFT: the modular curvature is directly related to the bulk Riemann curvature. We extend this approach by studying modular transport in the presence of a non-trivial quantum extremal surface. Focusing on JT gravity on an AdS background coupled to a non-gravitating bath, we compute the modular curvature of an interval in the bath in the presence of an island: the Quantum Extremal Modular Curvature (QEMC). We highlight some important properties of the QEMC, most importantly that it is non-local in general. In an OPE limit, the QEMC becomes local and probes the bulk Riemann curvature in regions with an island. Our work gives a new approach to probe physics behind horizons.

DOI
10.1007/jhep10(2024)006

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We show that the area operator of a quantum extremal surface can be reconstructed directly from boundary dynamics without reference to bulk geometry. Our approach combines the operator-algebra quantum error-correction (OAQEC) structure of AdS/CFT with modular Krylov complexity. Using Lanczos coefficients of boundary modular dynamics, we extract the spectrum of the modular Hamiltonian restricted to the algebra of the entanglement wedge and isolate its central contribution, which is identified with the area operator. The construction is purely boundary-based and applies to superpositions of semiclassical geometries as well. As an application, we diagnose island formation and the Page transition in evaporating black holes using boundary modular evolution alone, bypassing any bulk extremization. More broadly, our results establish modular Krylov complexity as a concrete and computable probe of emergent spacetime geometry, providing a new route to accessing black hole interiors from boundary quantum dynamics.
A bstract 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, 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.
Abstract We argue for an exponential bound characterizing the chaotic properties of modular Hamiltonian flow of QFT subsystems. In holographic theories, maximal modular chaos is reflected in the local Poincare symmetry about a Ryu-Takayanagi surface. Generators of null deformations of the bulk extremal surface map to modular scrambling modes — positive CFT operators saturating the bound — and their algebra probes the bulk Riemann curvature, clarifying the modular Berry curvature proposal of arXiv:1903.04493.