Quantum chaos in systems with mixed phase space

Unambiguously chaotic (globally hyperbolic) and integrable systems are separated by a gray zone that encompasses chaotic systems with mixed phase space. In such systems, “chaotic seas”, where trajectories diverge exponentially, coexist with “integrable islands” of periodic trajectories. In this talk, I will discuss how such systems behave after quantization and how to distinguish them from integrable systems with isolated saddle points. First, I will explain why fancy OTOCs are not suitable for working with “mixed” systems. Second, I will discuss how to correct this shortcoming using logarithmic and replica OTOCs, and analytically estimate the averaged Lyapunov exponent in the large-N limit. Finally, I will show that Krylov complexity correctly reproduces the local Lyapunov exponent in the semiclassical limit, given the correct choice of initial operator.