# Dynamics of quasiperiodically driven spin systems

## TL;DR

This paper investigates the complex dynamics of a spin system driven by Fibonacci-sequence kicks, revealing classical chaos with zero Lyapunov exponent and quantum fractality, along with ergodic thermalization when interacting with another spin.

## Contribution

It introduces a detailed analysis of Fibonacci-driven spin dynamics, deriving invariants, and contrasting classical chaos with quantum fractality and thermalization behaviors.

## Key findings

- Classical phase space shows chaos with zero Lyapunov exponent.
- Quantum states exhibit fractal fluctuations in spin operator means.
- Interaction induces ergodic dynamics and infinite temperature thermalization.

## Abstract

We study the stroboscopic dynamics of a spin-$S$ object subjected to $\delta$-function kicking in the transverse magnetic field which is generated following the Fibonacci sequence. The corresponding classical Hamiltonian map is constructed in the large spin limit, $S \rightarrow \infty$. Upon evolving such a map for large kicking strength and time period, the phase space appears to be chaotic; interestingly, however, the geodesic distance increases linearly with the stroboscopic time implying that the Lyapunov exponent is zero. We derive the Sutherland invariant for the underlying $SO(3)$ matrix governing the dynamics of classical spin variables and study the orbits for weak kicking strength. For the quantum dynamics, we observe that although the phase coherence of a state is retained throughout the time evolution, the fluctuations in the mean values of the spin operators exhibit fractality which is also present in the Floquet eigenstates. Interestingly, the presence of an interaction with another spin results in an ergodic dynamics leading to infinite temperature thermalization.