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Krylov complexity of many-body localization: Operator localization in Krylov basis

by Fabian Ballar Trigueros, Cheng-Ju Lin

Submission summary

As Contributors: Cheng-Ju Lin
Preprint link: scipost_202206_00018v1
Date accepted: 2022-08-02
Date submitted: 2022-06-20 05:56
Submitted by: Lin, Cheng-Ju
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as ∼ n/ ln(n) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.

Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Reports on this Submission

Anonymous Report 2 on 2022-7-28 (Invited Report)


The authors have addressed my concerns. The new draft is also more clearly presented. I am happy to recommend publication.

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Anonymous Report 1 on 2022-7-17 (Invited Report)


The authors have made a sincere effort at responding to all my questions. In addition, they have placed cautionary remarks regarding their extrapolation scheme, and the fact that their results hold for small system sizes. The various metrics used by the authors will serve as a guide for future studies. I therefore recommend publication.

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Anonymous on 2022-06-20  [id 2597]

Summary of changes.