Krylov complexity as an order parameter for quantum chaotic-integrable transitions

TL;DR

This paper introduces Krylov complexity peak as a universal order parameter for quantum chaotic systems, effectively distinguishing chaotic from integrable phases and correlating with traditional chaos indicators.

Contribution

It proposes the Krylov complexity peak as a new, operator-independent diagnostic for quantum chaos and demonstrates its effectiveness in identifying chaos-integrability transitions.

Findings

Krylov complexity peak correlates with quantum chaos indicators.

The peak height serves as an order parameter for chaos.

The method applies to models at finite and infinite temperature.

Abstract

Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems 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 chaotic systems and suggest that its height could serve as an `order parameter' for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs, while introducing an operator-independent diagnostic for quantum chaos, offering more `universal' insights and a deeper understanding of the general properties of quantum chaotic systems.