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Neural Network Quantum States analysis of the Shastry-Sutherland model

by Matěj Mezera, Jana Menšíková, Pavel Baláž, Martin Žonda

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Submission summary

Authors (as registered SciPost users): Pavel Baláž · Matej Mezera · Martin Žonda
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-12-04
Date submitted: 2023-11-03 11:48
Submitted by: Žonda, Martin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

Author comments upon resubmission

Dear Editor,

We hereby resubmit our manuscript for publication in SciPost Physics. The previous version was reviewed by three referees with an overall positive assessment of our work. Nevertheless, they also raised a number of questions, constructive criticism and useful recommendations. By reflecting on all reports of the referees and with the aim to further improve our manuscript we have prepared a new and extended version of the manuscript.

All changes are discussed in detail in the answers to the referees.

On behalf of all authors,
Martin Žonda

List of changes

By reflecting on all the questions, criticism and suggestions raised by the referees and with the aim to improve our manuscript we have made the following changes:

We have added benchmarks for J/J’ = 0.63 to Table 1 and briefly discuss these results in the main text.

We were able to converge the results for L=100 and, therefore, corrected the related data in the figure 5 and simplified the relevant text.

We have added iDMRG results taken graphically from the reference Phys. Rev. X 9(4), 041037 (2019) to the panel (e) in figure 5 and discuss the difference in the main text.

We have significantly expanded section 4.2.2 by an investigation of the plaquette phase for lattices with mixed boundary conditions (open and periodic). We have added a new composite figure 5. It shows a comparison of the RBM NQS results to DMRG taken from reference Phys. Rev. B 105(6), L060409 (2022) as well as the investigation of the position of the discontinuous phase transition between the dimer and plaquette phase. The results are discussed in length in the part “PS and mixed boundary conditions“. There we also address the strengths and weaknesses of the RBM NQS when compared to DMRG results for the SSM.

We have added a new appendix D in which we first show how RBM can encode dimer state and then demonstrate that it is expressive enough to encompass the plaquette ordering as well.

We have updated the conclusions to include the new results.

We have expanded the list of relevant references by works suggested by referees as well as by a few newly published works on the topic.

We corrected the wrong formats as well as spurious lower-casings in the list of references.

We have corrected typos, grammar mistakes, inconsistencies in referencing equations and figures and reformulated and simplified some sentences to make the text clearer.

A more detailed description of the changes can be found in our response to the referees.

Published as SciPost Phys. Core 6, 088 (2023)

Reports on this Submission

Anonymous Report 3 on 2023-11-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2303.14108v2, delivered 2023-11-25, doi: 10.21468/SciPost.Report.8188


I do appreciate the effort that the authors have put into revising their manuscript, in particular the additional results for $J/J'=0.63$ in Table 1, the new Appendix D, and the new figures 6 and 10. Nevertheless, there is still no new physics in the manuscript and from a technical point of view the present work relies on the NetKet libraries [9,56].

True, DMRG and iPEPS have a longer development history than neural network quantum states, but the present results are not only less accurate, but I do not see a major advance in the available toolbox either. Even if the present benchmarks remain interesting, this is in my opinion a clear case for SciPost Physics Core.

Requested changes

When rereading the manuscript, I noted a few details that the authors might wish to correct on the proofs:
1- I believe that the work that defined the name "Shastry-Sutherland model" should be cited in the first paragraph of the Introduction, and not just as Ref. [55] on page 5.
2- H. Kageyama et al., Phys. Rev. Lett. 82, 3168 (1999) is a central paper on SrCu$_2$(BO$_3$)$_2$ that has so far been overlooked. I recommend to cite it alongside / before Ref. [44].
3- There are three "dimmer"s in the manuscript (on pages 5, 19, and 25).
4- First paragraph of page 16: I believe that the correct mathematical term is "monotonically" (not "monotonously").
5- Top of page 23: the "differed ground state ordering" should probably read "different ground state ordering".
6- Ref. [59] seems to be published in PMLR 48:2990-2999, 2016 (albeit no DOI).

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

Anonymous Report 2 on 2023-11-18 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2303.14108v2, delivered 2023-11-18, doi: 10.21468/SciPost.Report.8145


The work opens up a new research direction and gives important guidelines for further improvements on simulating Shastry-Sutherland model with different architectures of neural network quantum states.


No new results on the Shastry-Sutherland model are reported.


The authors have answered all my question in great detail. In particular,
the representation of an exact dimer solid state has been addressed, and it has been argued that there may be issues in learning this state using the variational principle even though it is in the variational scope of the ansatz.
The authors have also tested how different types of boundary conditions that break the ground state degeneracy
affect the results in the plaquette singlet phase.

Given these improvements of the manuscript, I recommend publication in SciPost Physics
as the paper opens up a new research avenue by bridging the field of
neural quantum states with topical research on the
Shastry-Sutherland model, which is of great current interest due to direct experimental
realizations and which is hard to simulate numerically as only few methods can tackle this frustrated system.
One can expect that the authors' work will inspire and accelerate further studies.

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

Anonymous Report 1 on 2023-11-16 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2303.14108v2, delivered 2023-11-16, doi: 10.21468/SciPost.Report.8126


In the revised version the authors took into account referees' remarks. They increased the clarity of the manuscript. The authors provided quite an extensive overview of the neural network methods for quantum systems and then employed different schemes of them to study some ground-state phases of the Shastry-Sutherland model. The authors examine in detail the precision of the obtained results when changing the number of layers in the neural network (NN). One of the main messages is that a shallow NN can be sufficient to recover some phases of the model.

Unfortunately, the main question of the previous reviewer reports "what is the advantage of the methods employed in the manuscript with respect to other numerical methods" is not well elucidated. I understand that the current NQS methods could be computationally less demanding with respect to other methods, but on the other hand the results presented here concern the topics more or less well understood in the Shastry-Sutherland model. Therefore, no new knowledge of such a complex problem, as the Shastry-Sutherland model, has been gained and the current study does not meet the acceptance criteria for SciPost Physics.

Nevertheless, it is a nice paper which can be published in SciPost Physics Core.

Requested changes

Below Eq.(14):
The Planck constant is redundant in (-3/4)\hbar^2. The spin operators below Eq.(1) has been introduced without it.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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