We reveal a novel continuous dynamical heating transition between a prethermal and an infinitetemperature phase in a clean, chaotic periodically-driven classical spin chain. The transition time is a steep exponential function of the driving frequency, showing that the exponentially long-lived prethermal plateau, originally observed in quantum Floquet systems, survives the classical limit. Despite the inapplicability of Floquet’s theorem to nonlinear systems, we present strong evidence that the physics of the prethermal phase is described well by the inverse-frequency expansion, even though its stability and robustness are related to drive-induced coherence not captured by the expansion. Our results pave the way to transfer the ideas of Floquet engineering to classical many-body systems, and are directly relevant for cold atom experiments in the superfluid regime.

Periodically-driven systems are currently experiencing an unprecedented revival of interest through theoretical and experimental design of novel states of matter. Commonly known as Floquet engineering, this approach has enjoyed success in the regime of high driving frequency, where it has been appreciated as a useful tool to ascribe novel properties to otherwise trivial static Hamiltonians [1–3]. Prominent examples include the Kapitza pendulum [4], cold-atom realisations of topological [5–12] and spin-dependent [13] bands, artificial gauge fields [14–21], spin-orbit coupling [22, 23], enhanced magnetic correlations [24], synthetic dimensions [25–27], and photonic topological insulators [28–30]. The applicability of Floquet engineering requires the ability to prepare the periodically driven system in the corresponding Floquet state [31–33], and the stability of the system to detrimental heating [34, 35]. Presenting a major bottleneck at the forefront of present-date experimental research, heating processes play an important role in many-body Floquet systems, and understanding the underlying physics is expected to offer significant advances in the field. Unlike single-particle quantum systems, such as the kicked rotor [36] and bosonic band models [37, 38], it is commonly believed that generic isolated clean periodically-driven quantum many-body systems heat up to an infinite-temperature state [39–43]. Nonetheless, heating rates have been shown to be at least exponentially suppressed in the drive frequency [44, 45]. In this paper, we present a state-of-the-art numerical study of thermalisation in a clean, globally-driven, isolated classical spin chain, reaching times beyond the astronomical 1010 driving cycles. We find that, the dynamics falls into four stages, see Fig. 1: an initial transient (i) during which the system exhibits constrained ther-