Synchronization as a Route to Hierarchical Time-Crystalline Order
Synchronization is a well-established phenomenon in classical nonlinear oscillators. Time crystals[1,2], as intrinsically non-equilibrium phases that spontaneously break time translation symmetry, offer a natural setting to extend these ideas into the quantum regime. We explore synchronization and the role of seeding in stabilizing and controlling coherent many-body dynamics in time crystals [3,4]. We show that a time crystal coupled to melted counterparts can induce robust oscillations and uncover unconventional synchronization phenomena [3], including chimera states, and cluster synchronization [4]. We propose a new class of “Hierarchical time crystals” arising from the synchronization of discrete and continuous time crystals, leading to a richer symmetry-breaking structure and multiple emergent timescales [5]. Lastly, we discuss how higher-order synchronization within continuous-time crystals can reinforce this hierarchy, offering insights into engineering non-equilibrium phases.
[1] Rep. Prog. Phys. 81, 016401 (2018)
[2] Phys. Rev. Lett. 121, 035301 (2022)
[3] Phys. Rev. Lett. 128, 080603 (2022)
[4] Phys. Rev. Lett. 133, 260403 (2024)
[5] arXiv:2601.09779 (2026)
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