Assessing the saturation of Krylov complexity as a measure of chaos

TL;DR

This paper investigates whether Krylov complexity's long-time saturation reliably indicates chaos by analyzing its dependence on the choice of operator in an Ising chain model.

Contribution

It systematically studies the operator dependence of Krylov complexity saturation across the integrability-chaos transition.

Findings

Krylov complexity saturation varies significantly with different operators.

Its effectiveness as a chaos indicator depends on the specific operator chosen.

Comparison with spectral measures shows inconsistent correlation.

Abstract

Krylov complexity is a novel approach to study how an operator spreads over a specific basis. Recently, it has been stated that this quantity has a long-time saturation that depends on the amount of chaos in the system. Since this quantity not only depends on the Hamiltonian but also on the chosen operator, in this work we study the level of generality of this hypothesis by studying how the saturation value varies in the integrability to chaos transition when different operators are expanded. To do this, we work with an Ising chain with a transverse-longitudinal magnetic field and compare the saturation of the Krylov complexity with the standard spectral measure of quantum chaos. Our numerical results show that the usefulness of this quantity as a predictor of the chaoticity is strongly dependent on the chosen operator.