Universal chaotic dynamics from Krylov space

TL;DR

This paper explores how Krylov complexity and transition probabilities reveal universal features of quantum chaos, distinguishing chaotic from non-chaotic systems through spectral rigidity and wavefunction evolution.

Contribution

It derives an Ehrenfest theorem for Krylov complexity and identifies universal dynamical behaviors that characterize quantum chaos in Krylov space.

Findings

Chaotic systems show a universal rise-slope-ramp-plateau in transition probabilities.

A long ramp in transition probability indicates spectral rigidity and chaos.

Krylov complexity's late-time peak is linked to the ramp in transition probability.

Abstract

Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.