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Quantum quench dynamics in the transverse-field Ising model: A numerical expansion in linked rectangular clusters

by Jonas Richter, Tjark Heitmann, Robin Steinigeweg

Submission summary

As Contributors: Tjark Heitmann · Jonas Richter · Robin Steinigeweg
Arxiv Link: (pdf)
Date submitted: 2020-05-12
Submitted by: Richter, Jonas
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We study quantum quenches in the transverse-field Ising model defined on different lattice geometries such as chains, two- and three-leg ladders, and two-dimensional square lattices. Starting from fully polarized initial states, we consider the dynamics of the transverse and the longitudinal magnetization for quenches to weak, strong, and critical values of the transverse field. To this end, we rely on an efficient combination of numerical linked cluster expansions (NLCEs) and a forward propagation of pure states in real time. As a main result, we demonstrate that NLCEs comprising solely rectangular clusters provide a promising approach to study the real-time dynamics of two-dimensional quantum many-body systems directly in the thermodynamic limit. By comparing to existing data from the literature, we unveil that NLCEs yield converged results on time scales which are competitive to other state-of-the-art numerical methods.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 3 on 2020-6-16 Invited Report


State-of-the art numerical study of quantum quenches in spin models


The authors should clarify certain aspects about the methodology


The manuscript by Richter et al studies time evolution after quantum quenches in the transverse field Ising models in various lattice dimensions (chains, ladders and a square lattice). The main goal of the work is to compare two numerical techniques for unitary time evolution: the NLCE and exact diagonalization methods based on sparse matrices. The authors compare results case by case, and generally find that the NLCE is a competitive method that may provide in some parameter regimes very accurate predictions for time evolution of observables.

The manuscript is clearly written, the results are sound and of interest to a wide community of researchers studying nonequilibrium dynamics, quantum quenches, and developing computational methods. I therefore warmly recommend the paper for publication in SciPost Physics. It should be noted, though, that previous referees (reports 1 and 2) brought up relevant comments about the manuscript that should be answered before publication. The questions that I find particularly important are related to i) reasoning why rectangular clusters outperform other choices of clusters within the NLCE, ii) what are predictions of the NLCE if smaller orders of expansion are used, iii) to compare the NLCE results with exact diagonalization simulations with OBC.

Requested changes

Given by points i) - iii) in the report.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: good

Anonymous Report 2 on 2020-6-4 Invited Report


1-Potentially interesting method development for quantum dynamics in higher dimensions
2-Clearly written manuscript


1-The actual advantage of the method is not fully clear
2-Convergence of the cluster expansion does not seem to be easily controlled


The manuscript explores quench dynamics in the transverse field Ising model in a chain, ladders and 2D latices using a combination of linked cluster expansion and Chebyshev polynomial expansion for pure state evolution. The paper focuses on method development, the physics explored in the paper has been studied with other methods and therefore is used for comparison. Time evolution with linked cluster expansion has been studied before, even in this same model. The two new aspects of the method introduced in this paper is: i) selecting a subset of all possible clusters in the linked cluster expansion, and ii) using methods for the time evolution of a cluster that are more efficient than exact diagonalization, here they use the Chebyshev polynomial method.

The method the authors discuss is in principle interesting. The authors argue that some of their results show that one can go beyond exact diagonalization and obtain results at least comparable to other state-of-the-art methods for time evolution in 2D. While I think this is essentially true, I think the data shown in the paper suggest that the method is maybe not much more powerful than direct evaluation. Some aspects of the discussion are not fully clear as expanded on below.

I have a few direct comments on the manuscript and calculations:

1-On page 4 the authors write: "... the exact value of $g_c$ can vary due to finite-size corrections". Though I think I understand what the authors mean, the critical field by definition can not depend on finite-size corrections. Maybe the authors can clarify.

2-In Eq. (6) the authors introduce the multiplicities $\mathcal{L}_c$, but don't really define it. It would be useful for the reader if this was defined carefully.

3-Eq. (6) sums over "connected" clusters. While one may intuitively guess what is meant by connected here it would be beneficial to define this. In particular, I guess this depends on the Hamiltonian. For example, on the square lattice, connected sites are only nearest neighbours and not for example sites $(i,j)$ and $(i+1,j+1)$, right?

4-In several places the authors discuss topologically distinct clusters. I am confused about what they are talking about. As far as I can see, all clusters that the authors study in this paper are simply connected clusters in the plane. Therefore, in fact all clusters that they discuss are topologically equivalent. I did not see any representation of topologically distinct clusters (which would then have to be not simply connected). Maybe they mean something else with this phrase? This needs to be clarified.

5-On page 5, the authors write that "Eq. (6) can also converge for different types of expansions." This is an essential point of the paper. They have considered one type of expansion. Do the authors have any idea of what the principle for identifying useful approximate expansion is? Could they explain this? Or does one need to just work with trial and error?

6-Right after introducing the different expansion I mentioned in point 5, the authors mention the cluster with rectangular shape. But from what follows, it is only a restricted set of rectangular shapes that are used, at least in the ladders. Maybe the authors could clarify this in the text.

7-In Fig. 1 and below Eq. (8), the authors note that a cluster c and the same cluster rotated by 90 degrees, are topologically equivalent and therefore enter the sum (6) with multiplicity 2. It's not clear, partially since the multiplicity was not carefully defined, what this means. Do they simply mean that they can simply restrict the sum and multiply with two instead of summing over both types. I'm not sure I would say that this is equivalent to having multiplicity 2 in Eq. (6), since that equation would also have to be restricted for this to be true. Also, since this is in a section that is discussing the method in general, I would think that these two clusters are only equivalent if the Hamiltonian is actually invariant under rotations by 90 degrees. Is this true? Also here in this discussion, the notion of topologically equivalent clusters is a bit confusing, since all the clusters in Fig. 1 are topologically equivalent (not only the two green ones).

8-At the bottom of Eq. (7) the authors list the symmetries that are not present and could therefore not be used to speed up the calculation. But there is a parity symmetry that could be used. Why do they not mention this?

9-Moving on to results. In Fig. 2 the authors compare their method to time evolution on a finite system with PBC. At the same time, the clusters used in the linked cluster expansion have OBC. In principle these will have different finite size corrections, and in particular have different recurrence times. It would perhaps make more sense to compare with simulation on finite samples with OBC? Would it be possible to add this data? Also, the authors only show data for the linked cluster expansion with C = 24 and C = 25, that is, with clusters that are equally large as the finite size simulation. It would be interesting, to understand better how finite cluster effects appear, to also see data with cluster expansion with smaller C. Together these two, finite size with OBC and clusters with C < L, would better help make the point of the authors that "NLCE can yield a numerical advantage over the direct simulation of finite systems," which I don't think is fully clear from this data.

10-In the ladder simulation, why us there no data for g = 1 in Fig. 3b? Here, also, it would be interesting to see that finite system data with OBC.

11-In Fig. 5, it seems that the direct simulation of a 5x5 cluster is essentially equally good as the NLCE. Does that mean there is no advantage to the NLCE? Maybe a bit more discussion here would be useful.

12-In the comparison with ANN, it seems that the ANN data was stopped in time while still being accurate. Does that mean that the ANN could actually go to larger times and be comparable with the NLCE?

13-Do the authors have a way of knowing if the subset of clusters that they choose in their cluster expansion converges to a reliable result? One could for example imagine that within this subset the series Eq. (6) would converge and therefore only comparing different C within this subset would indicate convergence, but that then this convergence is to a value that is not the same as the full sum.

Requested changes

The requested changes are listed in the relevant points above.

  • validity: high
  • significance: ok
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2020-5-20 Invited Report


1) propose method to simulate quantum quenches
2) demonstrate that the method is competitive to other state-of-the-art methods
2) well-written, strengths and weaknesses are properly addressed


minor: missing some motivation for the method
minor: missing some motivation for particular choices in the algorithm


The authors propose to combine numerical linked cluster expansion with sparse time evolution algorithms to study the dynamics of quantum quenches. The algorithm is tested on a transverse-field Ising model in various geometries and compared to results from different methods including an exact solution for chains. For the two-dimensional square lattice limit, their method is compared to other recent methods and is demonstrated to be competitive.

The manuscript is well-written and the results are worked out in proper detail. The authors properly highlight the strengths and weaknesses of the method and the presentation is scientifically sound. Their results demonstrate that the proposed method is competitive with other state-of-the-art methods, including Artificial neural networks and iPEPS. There are, however, some questions I would like to address before the publication of the manuscript.

1) The study of quantum quenches is a broad and interesting research field. I would like the authors to elaborate more on the potential applications of the proposed method. What specific questions can this method answer, given current limitations in cluster size? Why are those questions relevant? Could the times that can be reached using their method be long enough to gain insights into certain physical phenomena? It would be good to give more explicit motivation here and realistic estimation of which scientific problems might be tackled.

2) I would like the authors to comment on the choice of rectangular subclusters in the numerical linked cluster expansion. I guess the choice is motivated by having to compute observables on fewer clusters than if arbitrary subclusters are considered. How does this change the convergence of the series in the linked-cluster expansion? Does one have to simulate larger clusters to reach the same accuracy than if all arbitrarily shaped subclusters are considered? There are some remarks in the section about the two-dimensional limit about this, but I would like the authors to elaborate on this a bit more.

3) The authors choose a Chebyshev polynomial approach to perform the time evolution. It would be good to have motivation for this choice. Does the Chebyshev polynomial approach have favorable properties in this context as opposed to other methods like Lanczos time evolution, or Runge-Kutta methods?

4) In Figure 2 it is not clear what the black boxes and circles are. It is explained in the text, but putting an explanation in the Figure caption would improve readability.

5) Figures 2, 3, 4 show how well the NLCE works when going to a large order of the cluster expansion. It would be interesting to see which times can be reached by which order of the expansion. Therefore, it would help if also a few results at lower orders are shown. Since this should now be new calculations, it would be great to add this data.

6) The authors mention that "exact diagonalization" is limiting the system size that allows for time evolution and therefore use the sparse Chebyshev algorithm to do so. I think that one could be more specific here, and call this "full exact diagonalization". Applying sparse algorithms such as Lanczos or Chebyshev algorithms on full wave functions is also often called "exact diagonalization", so this could be confusing. This is, of course, a very minor remark.

I think the manuscript would be of interest to many readers and the method will be useful in future studies. Upon addressing the remarks above, I can recommend the publication in SciPost.

Requested changes

see report

  • validity: top
  • significance: high
  • originality: good
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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