On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Continuing the previous initiatives arXiv: 2207.05347 and arXiv: 2212.06180, 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 gen...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Bhattacharya, Aranya, Nandy, Pratik, Pingal, Pratyush Nath, Sahu, Himanshu
Format: Article
Language:English
Subjects:
Algorithms
Complexity
Dissipation
Ising model
Iterative methods
Open systems
Subspaces
Online Access:Get full text
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Summary:Continuing the previous initiatives arXiv: 2207.05347 and arXiv: 2212.06180, 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 chaotic 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.
ISSN:2331-8422
DOI:10.48550/arxiv.2303.04175