Correlation Functions of Classical and Quantum Artin System defined on Lobachevsky Plane and Scrambling Time

TL;DR

This paper investigates the quantum properties of the classical Artin dynamical system on the Lobachevsky plane, demonstrating exponential decay of correlation functions and exponential growth of operator commutators, linking classical chaos to quantum scrambling.

Contribution

It provides the first detailed analysis of quantum correlation functions and scrambling behavior in the Artin system, connecting classical chaos with quantum thermal dynamics.

Findings

Quantum correlation functions decay exponentially with temperature-dependent exponents.

The square of the commutator grows exponentially, indicating quantum chaos.

Classical chaos influences quantum scrambling and operator growth.

Abstract

We consider the quantisation of the Artin dynamical system defined on the fundamental region of the modular group. In classical regime the geodesic flow in the fundamental region represents one of the most chaotic dynamical systems, it has mixing of all orders, Lebesgue spectrum and non-zero Kolmogorov entropy. As a result, the classical correlation functions decay exponentially. In order to investigate the influence of the classical chaotic behaviour on the quantum-mechanical properties of the Artin system we calculated the corresponding thermal quantum-mechanical correlation functions. It was conjectured by Maldacena, Shenker and Stanford that the classical chaos can be diagnosed in thermal quantum systems by using an out-of-time-order correlation function as well as the square of the commutator of operators separated in time. We demonstrated that the two- and four-point correlation functions of the Louiville-like operators decay exponentially with a temperature dependent exponent. As conjectured the square of the commutator of the Louiville-like operators separated in time grows exponentially, similar to the exponential divergency of trajectories in the classical regime. The corresponding exponent does not saturate the maximal growth condition.