Floquet engineering is a powerful manipulation method in modern quantum technology. However, unwanted heating remains a major challenge of Floquet engineering, motivating extensive studies of Floquet thermalization. In this work, we investigate the thermalization of periodically driven many-body systems through the lens of the Krylov chain, and uncover a topological origin of different thermalization behaviors. We show that the zero mode on the Krylov chain corresponds to a particular Floquet effective Hamiltonian with an unfolded spectrum for these driven systems. Furthermore, we demonstrate that when the Krylov chain has a finite band gap and nontrivial band topology, the driven system will thermalize to a finite-temperature state. If the Krylov chain is gapless, it will inevitably lead to infinite-temperature heating. We also reveal that Floquet prethermalization can be understood as the tunneling of a quasi edge mode across a local band gap with local nontrivial topology on the Krylov chain. This perspective provides a systematic route to constructing the effective prethermal Hamiltonian.

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• Nonequilibrium statistical mechanics
• Floquet systems
• Topological materials