Real-Time Quantum Dynamics on the Fuzzy Sphere: Chaos and Entanglement

TL;DR

This paper investigates the real-time quantum dynamics of a matrix model on the fuzzy sphere, analyzing chaos, entanglement, and thermal properties using Gaussian states and Lyapunov exponents.

Contribution

It introduces a Gaussian state approximation approach to study quantum chaos and entanglement in a fuzzy sphere matrix model, providing new insights into thermalization and scrambling.

Findings

Lyapunov exponent tends to zero at finite temperature, respecting the chaos bound.

High temperatures lead to classical chaotic behavior.

The model exhibits fast entanglement scrambling in real-time.

Abstract

We study the real time quantum dynamics of a matrix model consisting two bosonic fields on the fuzzy sphere using the Gaussian state approximation. Starting from the Hamiltonian formulation and using Wick's theorem, we derive a closed set of coupled nonlinear differential equations governing the time evolution of the one- and two-point correlation functions. Thermal equation of state is found by maximizing the von Neumann entropy over Gaussian states and solving algebraic self-consistency equation(s) leading to a complete determination of the symplectic spectrum of the covariance matrix. We identify near thermal initial conditions and use them to solve the equations of motion and employ our findings to probe chaos by calculating the largest Lyapunov exponent at various temperatures. Our results demonstrate that the latter tends to zero at a finite temperature indicating that the quantum dynamics respect the Maldacena,Shenker,Stanford bound across all temperatures, while approaching toward the classically chaotic regime at high temperatures. Finally, we examine the entanglement dynamics of the model in real-time by considering a sequence of bipartitions of the Hilbert space and computing the entanglement entropy and clearly exhibit the fast scrambling features that emerge in due detail.