Krylov Complexity in Mixed Phase Space

TL;DR

This paper explores Krylov complexity as a diagnostic tool for quantum chaos in systems with mixed phase space, revealing its correlation with eigenvalue statistics and its behavior across different models.

Contribution

It demonstrates that Krylov complexity reliably indicates quantum chaos and correlates with the Brody distribution in systems with mixed phase space, extending to various models.

Findings

Krylov complexity peaks in chaotic regimes.

It correlates with the Brody distribution parameter.

Complexity diminishes as systems become more integrable.

Abstract

We investigate the Krylov complexity of thermofield double states in systems with mixed phase space, uncovering a direct correlation with the Brody distribution, which interpolates between Poisson and Wigner statistics. Our analysis spans two-dimensional random matrix models featuring (I) GOE-Poisson and (II) GUE-Poisson transitions and extends to higher-dimensional cases, including a stringy matrix model (GOE-Poisson) and the mass-deformed SYK model (GUE-Poisson). Krylov complexity consistently emerges as a reliable marker of quantum chaos, displaying a characteristic peak in the chaotic regime that gradually diminishes as the Brody parameter approaches zero, signaling a shift toward integrability. These results establish Krylov complexity as a powerful diagnostic of quantum chaos and highlight its interplay with eigenvalue statistics in mixed phase systems.