Fractal nature of high-order time crystal phases

In Floquet time crystals, characterized by the spontaneous breaking of time-translational invariance, observables exhibit oscillations with a period multiple $p>2$ of the driving. Here, a new, experimentally accessible, order parameter is introduced, able to detect time-crystalline phases regardless of the value of p. This unveils a rich landscape with self-similar fractal boundaries in the phase diagram of the long-range kicked Ising model. These features are explained within the theoretical framework, along with the emergence of the Zₚ symmetry in Bloch eigenstates.