Integrability to chaos transition through Krylov approach for state evolution

TL;DR

This paper explores the transition from integrability to chaos in quantum systems using the Krylov basis approach, demonstrating that Krylov complexity and Lanczos coefficient dispersion can indicate quantum chaos depending on initial conditions.

Contribution

It introduces a Krylov-based method to analyze quantum chaos transitions, highlighting the importance of initial states in measuring chaos through Krylov complexity and Lanczos coefficients.

Findings

Krylov complexity saturation depends on initial conditions.

Lanczos coefficient dispersion varies with initial states.

Both metrics can effectively gauge quantum chaos with proper initial state selection.

Abstract

The complexity of quantum evolutions can be understood by examining their dispersion in a chosen basis. Recent research has stressed the fact that the Krylov basis is particularly adept at minimizing this dispersion [V. Balasubramanian et al, Physical Review D 106, 046007 (2022)]. This property assigns a central role to the Krylov basis in the investigation of quantum chaos. Here, we delve into the transition from integrability to chaos using the Krylov approach, employing an Ising spin chain and a banded random matrix model as our testing models. Our findings indicate that both the saturation of Krylov complexity and the dispersion of the Lanczos coefficients can exhibit a significant dependence on the initial condition. However, both quantities can gauge dynamical quantum chaos with a proper choice of the initial state.