Chaotic-integrable transition and Mixed Phase with Krylov complexity

Krylov complexity has recently emerged as a new paradigm for characterizing quantum chaos in many-body systems. Recent studies have shown that, in quantum chaotic systems, the Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work, we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaos and suggest that its height can serve as an effective order parameter. We investigate the Krylov complexity of thermofield double states in systems with mixed phase space, revealing a clear correlation with the Brody distribution, which interpolates between Poisson and Wigner level statistics. Our analysis spans two-dimensional random matrix models featuring (i) GOE-Poisson and (ii) GUE-Poisson transitions, and extends to higher-dimensional systems, including a stringy matrix model (GOE-Poisson) and the mass-deformed SYK model (GUE-Poisson). These results establish Krylov complexity as a powerful diagnostic of quantum chaos, highlight its interplay with level statistics in mixed phase systems, and offer deeper insights into the general properties of quantum chaotic systems.