# Many-body synchronisation in a classical Hamiltonian system

## TL;DR

This paper investigates a phase transition in a classical Hamiltonian system where many interacting spins synchronize under periodic driving, revealing a novel form of time crystal behavior without dissipation.

## Contribution

It demonstrates the existence of a synchronization transition in a purely Hamiltonian system, connecting it to time crystal phenomena and expanding understanding of collective dynamics.

## Key findings

- Synchronization occurs in a Hamiltonian system without dissipation.
- A transition between synchronized and chaotic regimes is identified.
- Synchronization can happen in both regular and chaotic overall dynamics.

## Abstract

We study synchronisation between periodically driven, interacting classical spins undergoing a Hamiltonian dynamics. In the thermodynamic limit there is a transition between a regime where all the spins oscillate synchronously for an infinite time with a period twice as the driving period (synchronized regime) and a regime where the oscillations die after a finite transient (chaotic regime). We emphasize the peculiarity of our result, having been synchronisation observed so far only in driven-dissipative systems. We discuss how our findings can be interpreted as a period-doubling time crystal and we show that synchronisation can appear both for an overall regular and an overall chaotic dynamics.