The quantum cat map on the modular discretization of extremal black hole horizons

TL;DR

This paper models the chaotic quantum evolution near extremal black hole horizons using a quantum Arnol'd cat map, demonstrating fast thermalization and ETH compliance in a discretized AdS$_2$ geometry.

Contribution

It introduces a toy model employing the quantum Arnol'd cat map for black hole horizon dynamics, explicitly constructing eigenstates and analyzing thermalization.

Findings

Eigenstates are random and satisfy ETH.

Thermalization is exponentially fast, saturating the STB.

Finite Hilbert space dimension is restricted to Fibonacci integers.

Abstract

Based on our recent work on the discretization of the radial AdS$_2$ geometry of extremal BH horizons,we present a toy model for the chaotic unitary evolution of infalling single particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single particle dynamics for an observer falling into the BH horizon, with time evolution operator the quantum Arnol'd cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single--particle Hilbert space takes values in the set of Fibonacci integers.