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Krylov complexity in a natural basis for the Schrödinger algebra
by Dimitrios Patramanis, Watse Sybesma
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Watse Sybesma |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2306.03133v4 (pdf) |
| Date accepted: | 2024-05-28 |
| Date submitted: | 2024-04-10 20:56 |
| Submitted by: | Sybesma, Watse |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate operator growth in quantum systems with two-dimensional Schr\"odinger group symmetry by studying the Krylov complexity. While feasible for semisimple Lie algebras, cases such as the Schr\"odinger algebra which is characterized by a semi-direct sum structure are complicated. We propose to compute Krylov complexity for this algebra in a natural orthonormal basis, which produces a pentadiagonal structure of the time evolution operator, contrasting the usual tridiagonal Lanczos algorithm outcome. The resulting complexity behaves as expected. We advocate that this approach can provide insights to other non-semisimple algebras.
Author comments upon resubmission
Published as SciPost Phys. Core 7, 037 (2024)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2024-5-22 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2306.03133v4, delivered 2024-05-22, doi: 10.21468/SciPost.Report.9113
Strengths
gives new explicit calculations of Krylov complexities by exploiting a symmetry-based approach.
Weaknesses
1. The scope of the results limited
2. Only limited physical interpretation given by authors
Report
Krylov complexity is a relatively recent measure of quantum complexity, whose basic phenomenology is being actively investigated. The present article adds to this field of study by extending the formalism to theories with Schrödinger group symmetry, and argue that their method more generally lets them compute Krylov complexity for symmetry groups that posses a semi-direct sum structure.
The use of the so-called “natural basis” in which the Liouvillian is penta-diagronal is a departure from the usual Krylov algorthithm, although the authors argue that the same Krylov space is still probed.
I find the results interesting and the responses and changes made in response to another referee’s comments to have improved the paper sufficiently to merit publication. Due to the somewhat limited scope of the analysis I would, however, recommend publication in SciPost Core.
Recommendation
Accept in alternative Journal (see Report)