Krylov localization in periodically driven quantum systems

Krylov construction reduces involved state or operator dynamics to a simple one-dimensional problem, making it an indispensable tool for exploration of complex quantum systems. We generalize this construction to periodically driven (Floquet) systems using the theory of orthogonal polynomials on the unit circle. Compared to Arnoldi iteration, our method works faster and maps quantum dynamics into a one-dimensional tight-binding model. We show that hopping parameters of this model are related to the energy level density, which implies that Krylov wave function is localized in integrable systems and delocalized in chaotic systems. Moreover, we show that Krylov localization length captures mixing properties of a quantum system in the semiclassial limit. These observations justify the use of Krylov delocalization as a hallmark of chaos in quantum Floquet systems.