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.
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Acknowledgments
We are grateful to Vijay Balasubramanian, Souvik Banerjee, Pawel Caputa, Adolfo del Campo, Pratik Nandy, and Dario Rosa for discussions. The research of J. E. and Z.-Y. X. is funded by DFG through the Collaborative Research Center SFB 1170 ToCoTronics, Project-ID 258499086—SFB 1170, as well as by Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat (EXC 2147, project-id 390858490). Z .-Y. X. also acknowledges support from the National Natural Science Foundation of China under Grants No. 11875053 and No. 12075298. S.-K.J is supported by a startup fund at Tulane University.
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Erdmenger, J., Jian, SK. & Xian, ZY. Universal chaotic dynamics from Krylov space. J. High Energ. Phys. 2023, 176 (2023). https://doi.org/10.1007/JHEP08(2023)176
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DOI: https://doi.org/10.1007/JHEP08(2023)176