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Thermalization of long range Ising model in different dynamical regimes: a full counting statistics approach
by Nishan Ranabhat, Mario Collura
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Submission summary
| Authors (as registered SciPost users): | Mario Collura · Nishan Ranabhat |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.00533v2 (pdf) |
| Date submitted: | 2023-01-27 11:10 |
| Submitted by: | Ranabhat, Nishan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We study thermalization of transverse field Ising chain with power law decaying interaction $\sim 1/r^{\alpha}$ following a global quantum quench of the transverse field to two different dynamical regimes. We quantify the thermalization behavior by comparing the full probability distribution function (PDF) of the evolving states with the corresponding thermal state given by the Gibbs canonical ensemble (GCE). To this end, we use matrix product state (MPS) based time dependent variational principle (TDVP) algorithm to simulate both real time evolution following a global quantum quench and the finite temperature density operator. We observe that thermalization is strongly suppressed in the region with strong confinement for all the interaction strength $\alpha$ considered whereas thermalization occurs in the region with weak confinement.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-3-11 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.00533v2, delivered 2023-03-11, doi: 10.21468/SciPost.Report.6881
Strengths
1 interesting idea to study thermalization using the full counting statistics
2 suitable and interesting choice of systems, making it possible to study thermalization against dynamical confinement in long-range interacting systems
Weaknesses
1 Unclear, how central quantities (in particular Eq (8), the central quantity studied in the paper) are computed using the matrix-product state techniques applied by the authors
2 Unclear, how valid the numerical results are: MPS+TDVP simulations can lead to qualitatively wrong results for certain initial states, a necessary test is not discussed in the manuscript. Also, the accuracy in particular at long times is determined by the discarded weight, which is not mentioned in the manuscript.
3 English grammar can be improved throughout the manuscript
Report
The manuscript addresses the question for thermalization in long-range ferromagnetic transverse field Ising models using numerical simulations of the time evolution obtained via matrix product states (MPS). The main quantity computed is the full probability distribution function (PDF), which is a very interesting quantity to study for the thermalization dynamics, since it allows one to learn more than by simply looking at the time evolution of a local observable. The results - provided the numerics is not erroneous, see below - are interesting and should be published. However, I feel that SciPost Physics Core would be a more appropriate journal. The following points are important to be addressed by the authors prior to publication:
- it is not clear to me, how the main quantity of their study, the PDF in Eq. (8) is computed using MPS. The reader would benefit from further explaining how this is done using the MPS-approach. If this is too technical, it can be added to the appendix.
- the results look sound; however, from own experience, MPS+TDVP can lead to qualitatively wrong results for certain initial states, in particular product states. Here, the initial state does carry entanglement, which should help the simulations, but the initial bond dimension of 2 is still very small. It is necessary that the authors compare to exact diagonalization results for small systems to make sure the TDVP is not running in a wrong direction for the cases studied. This should be added either in the main discussion of the results, or in the appendix.
- even though not very likely, also finite-temperature results can get stuck or go to wrong results. Again, the initial product state might lead to problems, in particular in connection with the long-range interactions. It is easy to check the accuracy of their results by computing , e.g., the energy as a function of temperature using exact diagonalizations (e.g., using QuSpin) and show a meaningful example in the manuscript (main text or appendix).
- it is not clear, how big their numerical error is. This can be estimated by computing the discarded weight as a function of time (or inverse temperature, resp.), and the authors should provide typical / maximal values of the discarded weight in their simulations. Ideally, comparison to results with other bond dimensions should be shown.
Requested changes
1 provide more details on how calculations were performed, in particular Eq. (8) using MPS (this can be added in the appendix, if too technical).
2 give more insights into accuracy of the numerical results, e.g. by providing the maximum discarded weight encountered in their simulations.
3 benchmark the MPS+TDVP time-evolutions for the given initial state and Hamiltonians for small systems against exact diagonalizations, since TDVP for this initial state might lead to erroneous results (can be included in the appendix).
4 benchmark the finite temperature results against exact diagonalizations (only a generic test for a typical system suffices, can be added to the appendix).
Report #1 by Anonymous (Referee 3) on 2023-3-10 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2212.00533v2, delivered 2023-03-10, doi: 10.21468/SciPost.Report.6834
Report
In this work the authors study thermalization in two different dynamical regimes of the long range transverse field Ising model after a global quench. To determine whether or not the system is thermalizing they follow the time evolution of the distance of the system to the thermal state. Since this approach relies on the full probability distribution function rather than the expectation value of one or several observables, which is often the case in other studies, it should provide better results. However, as the authors themselves note, this is not the first time this approach has been used. Thus this is a case of applying an already established method to this particular model. Furthermore, the results also seem to mostly confirm what has previously been known from other studies. The only possible exception here is the observation that the metric considered in this paper shows thermalization at weak confinement. However, this is left unexplored. Based on this, I do not believe that the paper, in its current form, satisfies SciPosts acceptance criteria and additional work should be put towards understanding the relation between confinement and thermalization before considering the paper for publication.
Regarding the presentation of the paper, it leaves a lot to be desired. Not only does the text contain many grammatical errors, but the main results of the work are not conveyed very clearly. I would recommend the authors more clearly emphasize the novel results presented in this work. Additionally, a large part of the text is assigned to describing TDVP. While this would typically be relegated to the Appendix I see no issue with this, however, it may be useful to do something similar for the model. Namely, the authors often refer to the different dynamical regimes present here. Perhaps a few sentences could be added briefly describing the previous works and what is already known about these regimes. This would also make the point of the current work clearer to non-experts. Finally, it would also be beneficial if Figures 2 and 3 were moved to the main text, where they are referenced.
Moving on, this work relies exclusively on numerical simulations of finite systems using TDVP. While the authors do provide plots comparing different system size to show that their results are not significantly affected by finite size effects (at least at the larger system sizes L=60 and L=100), the authors do not provide any evidence for the convergence of their results with bond dimension $\chi$. While there is no reason to believe the authors did not check this internally it would be beneficial to provide some data showing the convergence with bond dimension.
Additionally, I am wondering why the authors limited their study to only two values of the magnetic field. In this way they verify that the method distinguishes between two dynamical regimes. However, it would be much more interesting to see whether the method can also capture the transition between these regimes in the thermodynamic limit. Furthermore, looking at different values one might be able to understand why "strong" confinement leads to a lack of thermalization while "weak" confinement does not. Is it simply that the case of "weak" confinement is not really confined at all, but only seems to be confined due to short timescales? I believe some of these questions should be explored before the paper can be considered for publication.
Finally, a few minor issues:
- In the abstract the authors mention the Gibbs cannonical ensemble abbreviated as GCE, however, throughout the paper you switch to cannonical Gibbs ensemble CGE. Perhaps this could be made consistent.
- In the main text the authors reference figures 1, 2, 3 and then 11. Since Figure 11 is not part of the main text, perhaps that reference should be replaced with the reference to the relevant Appendix.
- Regarding the Figures, I would recommend reconsidering the use of 3D plots in this case as they simply make it more difficult to see the main results.
- In Figure 3 the Authors claim that there is a significant difference in panels b) and c). I find it difficult to agree with that statement based on what is shown in panels b) and c) alone. Any relevant difference that might be present would only be visible clearly using a log scale.