18 May 2023 Thu 5 pm

The Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. There has been a long-standing debate regarding the relationship between the growth of Krylov complexity and the chaotic nature of the Hamiltonian. In this study, we investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. 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 a Krylov basis, which is a characteristic of chaos in the spectrum of the Hamiltonian. Additionally, the long ramp in the transition probability is directly responsible for the late-time peak of Krylov complexity observed in previous literature. On the other hand, for non-chaotic systems, the transition probability exhibits a different behavior without the long ramp. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos.

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