Abstract
Continuing the previous initiatives [1, 2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and non-integrable systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.
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Acknowledgments
We wish to thank Xiangyu Cao, Rathindra Nath Das, Bidyut Dey, Hosho Katsura, Amin Nizami, Tanay Pathak, Shinsei Ryu, Aninda Sinha, Julian Sonner, and Hironobu Yoshida for useful discussions in related works. We also thank the anonymous referee for valuable suggestions, which leads to the subsection 3.2.3. A.B. thanks the organizers of the workshop on “QI in QFT and AdS/CFT III” and the hospitality of the Theory Division, Saha Institute of Nuclear Physics, where part of this work was presented. P.N. would like to thank Princeton Center for Theoretical Science (PCTS), Princeton University for hosting him through the overseas visiting program for young researchers (KAKENHI No. 21H05182) during the final stages of the work. The work of A.B. is supported by the Polish National Science Centre (NCN) grant 2021/42/E/ST2/00234. 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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Bhattacharya, A., Nandy, P., Nath, P.P. et al. On Krylov complexity in open systems: an approach via bi-Lanczos algorithm. J. High Energ. Phys. 2023, 66 (2023). https://doi.org/10.1007/JHEP12(2023)066
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DOI: https://doi.org/10.1007/JHEP12(2023)066