Abstract
We use Krylov complexity to study operator growth in the q-body dissipative Sachdev-Ye-Kitaev (SYK) model, where the dissipation is modeled by linear and random p-body Lindblad operators. In the large q limit, we analytically establish the linear growth of two sets of coefficients for any generic jump operators. We numerically verify this by implementing the bi-Lanczos algorithm, which transforms the Lindbladian into a pure tridiagonal form. We find that the Krylov complexity saturates inversely with the dissipation strength, while the dissipative timescale grows logarithmically. This is akin to the behavior of other 𝔮-complexity measures, namely out-of-time-order correlator (OTOC) and operator size, which we also demonstrate. We connect these observations to continuous quantum measurement processes. We further investigate the pole structure of a generic auto-correlation and the high-frequency behavior of the spectral function in the presence of dissipation, thereby revealing a general principle for operator growth in dissipative quantum chaotic systems.
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Acknowledgments
We would like to thank Alexei Andreanov, Aranya Bhattacharya, Xiangyu Cao, Saskia Demulder, Anatoly Dymarksy, Damián A. Galante, Hosho Katsura, Tokiro Numasawa, Dario Rosa, Shinsei Ryu, Masaki Tezuka, and Hironobu Yoshida for discussions on related topics. Numerical calculations were performed in the workstation Octaquark-4, CHEP, and using the computational facilities of YITP. P.N. acknowledges the hospitality of the Indian Institute of Science, Saha Institute of Nuclear Physics, The University of Tokyo during the final stages of the work and the long-term workshop YITP-T-23-01 held at YITP, Kyoto University, where some of the results were presented. T.P. thanks the generous hospitality of the Department of Physics, Kyoto University. B.B. acknowledges the financial support from the Institute for Basic Science (IBS) through the project IBS-R024-D1. The work of P.N. is supported by the JSPS Grant-in-Aid for Transformative Research Areas (A) “Extreme Universe” No. 21H05190.
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Bhattacharjee, B., Nandy, P. & Pathak, T. Operator dynamics in Lindbladian SYK: a Krylov complexity perspective. J. High Energ. Phys. 2024, 94 (2024). https://doi.org/10.1007/JHEP01(2024)094
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DOI: https://doi.org/10.1007/JHEP01(2024)094