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Krylov complexity of many-body localization: Operator localization in Krylov basis
by Fabian Ballar Trigueros, Cheng-Ju Lin
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|Authors (as Contributors):||Cheng-Ju Lin|
|Arxiv Link:||https://arxiv.org/abs/2112.04722v1 (pdf)|
|Date submitted:||2022-01-10 14:02|
|Submitted by:||Lin, Cheng-Ju|
|Submitted to:||SciPost Physics|
Motivated by the recent developments of quantum many-body chaos, characterizing how an operator grows and its complexity in the Heisenberg picture has attracted a lot of attentions. In this work, we study the operator growth problem in a many-body localization (MBL) system from a recently proposed 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 phase, the Lanczos coefficients scales $\sim n/ \ln(n)$ asymptotically, same as in the ergodic phase, but with an additional even-odd alteration and effective randomness. Extrapolating the Lanczos coefficients to the thermodynamic limit, we study the spectral function and also find that the corresponding single-particle problem is localized for both unextrapolated and extrapolated Lanczos coefficients, resulting in a bounded Krylov complexity in time. For the MBL phenomenological model, the Lanczos coefficients also have an even-odd alteration, but approaching to constants asymptotically. We also find that the Krylov complexity grows linearly in time for the MBL phenomenological model.
Submission & Refereeing History
- Report 2 submitted on 2022-07-28 23:06 by Anonymous
- Report 1 submitted on 2022-07-17 22:51 by Anonymous
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Reports on this Submission
Anonymous Report 2 on 2022-3-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.04722v1, delivered 2022-03-19, doi: 10.21468/SciPost.Report.4729
The topic of the paper: a study of systems with disorder and interactions in general, and many body localized (MBL) systems in particular, is very interesting and timely. Moreover the use of the Krylov method to the study of these systems, is a promising endeavor.
1. Poor citation of past literature which has discussed the even-odd effect and its influence on operator spreading.
2. Unconvincing interpretation of the numerical results.
The paper studies MBL systems using the method of mapping operator dynamics to single particle dynamics in a Krylov subspace. By studying the hopping parameters bn of the Krylov chain, the authors argue that while the overall behavior of the bn are very similar to chaotic systems by being n/ln(n), the bn also show an even odd effect which is responsible for localizing the particle in Krylov subspace. I have the following comments:
1. The effect of the even-odd effect on localization is not new. The authors should make this fact clear when discussing past literature. The book by Vishwanath that the authors cite, discusses even-odd effects. The two papers by Yates et al that the authors cite, also discuss even odd effects in detail.
2. The papers by Yates et al showed that when there is an even odd effect superimposed on a linear slope, the particle on the first site of the Krylov chain, stays localized. However if the even-odd effect persists only up to a finite distance into the chain, it makes the operator quasi-stable, causing the operator to decay into the bulk. These decay times can be very long. Looking at the authors plots, it appears they have a quasi-stable mode at the end of the Krylov chain rather than a stable mode as their even-odd effect does not persist into the bulk. Moreover the point where the even-odd effects terminate appears to be system size independent. So I am not convinced that their results imply a delta function peak in their low frequency spectral weight. The authors should discuss this physics and consider a more careful interpretation of their results.
This is also important as recent papers by Sels et al and Vidmar et al, have raised questions about past numerical metrics for MBL and there is new debate again on absolute stability vs quasi-stability.
In summary the paper presents solid numerical data, but I have concerns about the physical interpretation.
1. Proper citation of past literature in the context of the even-odd effect in Krylov chains.
2. A more careful discussion of their numerical results especially concerning the finite extent of the even-odd effects in their Krylov chain.
3. A more up-to-date citation to numerical studies of MBL such as recent studies by Sels et al and Vidmar et al.
Anonymous Report 1 on 2022-2-20 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2112.04722v1, delivered 2022-02-20, doi: 10.21468/SciPost.Report.4478
1. The authors study an interesting and open question in the subfield.
1. Some of the central claims do no have sufficiently detailed numerical support.
2. There are more typos than there should be.
This work investigates the Operator Krylov Method (OKM) -- specifically a recent formulation of it in -- in many-body localized systems. It's an interesting question, as OKM makes the assumption that operator lanczos coefficients grow linearly asymptotically. While this assumption appears to hold for ergodic systems, it's unclear how it is modified in the presence of disorder, and therefore it's unclear whether OKM is useful in these systems.  suggests that MBL appears as a subleading correction to the asymptotic form of the Lanczos coefficients.
The present paper investigates this issue further. They appear to find (numerically) that the lanczos coefficients (while growing linearly) have an even-odd modulation as well as an O(1) random fluctuation. This leads to an interesting conclusion: When the Liouvillian is expressed in the Lanczos basis, it ends up resembling a disordered single-particle hopping problem. The authors suggest that MBL corresponds to localization in this single particle problem, which is indeed an interesting suggestion.
While this manuscript likely meets the acceptance criteria (groundbreaking theoretical discovery), the authors will have to do more work to substantiate their main numerical claims.
I'll discuss the major physics issues here, and list typos/smaller issues in the requested changes.
First, the central claim (MBL corresponds to localization in Lanczos basis). The central numerical evidence for this claim is Fig 5-7. While these figures are quite compelling, it's unclear how well they really support the underlying hypothesis. For example, to be sure that the plateau in fig 6 correspond to localization, we need to see that the source of the plateau isn't simply the smallness of the system size. The authors should make clear the system sizes used for each figure, and also report how the results change with system size (perhaps in an appendix).
Second, the authors report that in the phenomenological model, lanczos coefficients saturate. As it stands this is a somewhat unilluminating part of the work. Do the lanczos coefficients saturate because the authors initial operator is an l-bit? Or does the lanczos coefficients saturate because the phenomenological model is limited (and only includes 2 body lbit operators)?
1. Report system sizes in all figure captions.
2. Provide data on convergence with system size in fig 6 at least, and add a discussion of this point to the text.
3. Clarify reason for differences between phenomenological model and MBL model (see report).
4. The last paragraph of the conclusion is very difficult to follow. The authors should clarify what they mean, and provide examples in the literature of "using computation complexity to characterise different phases of matter".
5. Top of page 3: "to mention" -> "mentioning"
6. Caption fig 5: "Noe" -> "Now"
7. pg 9 "unextrpolated" -> "unextrapolated"