Krylov Complexity as a Probe for Chaos

TL;DR

This paper investigates how the time evolution of Krylov complexity can distinguish between chaotic and integrable many-body systems, revealing characteristic saturation behaviors that serve as chaos indicators.

Contribution

It analytically and numerically demonstrates that the saturation pattern of Krylov complexity uniquely identifies chaos versus integrability in quantum systems.

Findings

Chaotic systems reach saturation complexity at finite times with possible peaks.

Integrable systems approach saturation more slowly from below.

Saturation behavior effectively probes system chaos.

Abstract

In this work, we explore in detail, the time evolution of Krylov complexity. We demonstrate, through analytical computations, that in finite many-body systems, while ramp and plateau are two generic features of Krylov complexity, the manner in which complexity saturates reveals the chaotic nature of the system. In particular, we show that the dynamics towards saturation precisely distinguish between chaotic and integrable systems. For chaotic models, the saturation value of complexity reaches its infinite time average at a finite saturation time. In this case, depending on the initial state, it may also exhibit a peak before saturation. In contrast, in integrable models, complexity approaches the infinite time average value from below at a much longer timescale. We confirm this distinction using numerical results for specific spin models.