Neural Network Quantum States analysis of the Shastry-Sutherland model

We would like to thank the referee for the constructive criticism as well as for the recommendation to publish our work as a regular paper in SciPost Physics Core. Nevertheless, we would still like to address all raised points in detail and discuss how we have improved our manuscript reflecting on the report. Hopefully, this will convince the referee that the revised version of our manuscript is worth publishing in SciPost Physics as well.

Another Referee already pointed out that one might have cited further references. Even if I admit that there are so many investigations of the Shastry-Sutherland model that a complete overview is not possible, let me nevertheless mention the work by M. Jaime et al., PNAS 109, 12404 (2012) that one could have mentioned alongside Ref. [31].

We have added the requested reference, as well others relevant referees to the overview part of the paper.

While this is overall a nice benchmark with a large review component, I am a bit concerned about the novelty of the work. Specifically, the ED study Ref. [29] already investigated ground-state properties for significantly bigger systems than those investigated here, not to mention the iPEPS study Ref. [32]. Likewise, magnetization curves were already carefully investigated previously, e.g., in Refs. [31,33] and the PNAS by Jaime et al. It is true that here the authors make an effort to go to bigger lattices (N≤64), but they focus on a parameter regime that is deeper in the dimer-singlet phase than the regime J/J′≈0.63 relevant to SrCu2(BO3)2. Thus, there is no new physics. Even more, I understand that also the methods have been introduced elsewhere such that the only new element of the present work is the benchmark on a different model.

We understand the concerns of the referee about the scientific novelty of the paper and, as discussed below, we have now made an effort to boost the paper in this respect. However, we would also like to point out to the two other possible route for publication in SciPost Physics listed on the webpage as well as in the report of the referee: Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work; Provide a novel and synergetic link between different research areas. We have shown in our paper that the Shastry-Sutherland model is not just another model on which the NQS approach can be applied. It is a model exceptionally suited for the task. This is partially because there are so many exact (even analytical) or close to exact (e.g., iDMRG) results obtained for this model as mentioned by the referee. But it is mainly because it has a complicated phase structure. There are continuous as well as discontinuous phase transitions, novel quantum phases, plateaus in the magnetizations, several open problems with conflicting results from different studies and the list goes on. We think that we are therefore opening a new pathway in existing research. We also combine two different research areas, namely the research of Shastry-Sutherland systems and NQS development.

The new elements are not just the benchmarks. First of all, we show that complicated networks are not necessarily the right answer to all problems. Then we introduced several strategies for using simple NQS on such a complicated system as the Shastry-Sutherland model. We have now demonstrated that a simple RBM can capture dimer state, AF state and also plaquette state, although the last one is difficult to learn. We discuss the advantages and disadvantages of transfer learning. We have also added some benchmarks for J/J′≈0.63 (see Table 1) relevant to SrCu2(BO3)2 at normal pressure. We have shown that NQS can deal with the step-like structure of the magnetisation plateaus. To our knowledge, this is a qualitatively novel result opening a new possible application of the method. For example, there are very recent results discussing a possibility of spin-supersolid orderings at finite magnetic fields [Nat Commun 14, 3769 (2023)] that could be now addressed with NQS. Further, we have now pushed the even larger lattices and also discussed mixed boundary conditions. We showed that this leads to a more realistic estimation of the position of the discontinuous phase transition, although the difficulties to learn PS prevail. To address this issue, we have introduced another strategy in the updated version of the manuscript to deal with this regime more efficiently. We also added comparison for periodic and mixed boundary conditions with DMRG and iDMRG results and discussed some problems this revealed.

More on the level of detail, I believe that NQS and specifically RBM are reminiscent of tensor networks. Accordingly, I think that at least some comparison to previous tensor network and specifically iPEPS investigations would be appropriate, beyond just mentioning the existence of these works.

There is definitely a connection between NQS and tensor networks and it was already investigated in the literature. See for example: D.-L. Deng, X. Li, and S. Das Sarma, Quantum entanglement in neural network states, Phys. Rev. X 7, 021021 (2017). J. Chen, S. Cheng, H. Xie, L. Wang, and T. Xiang, Equivalence of restricted boltzmann machines and tensor network states, Phys. Rev. B 97, 085104 (2018). Y. Levine, O. Sharir, N. Cohen, and A. Shashua, Quantum Entanglement in Deep Learning Architectures, Phys. Rev. Lett. 122, 065301 (2019). L. Pastori, R. Kaubruegger, and J. C. Budich, Generalized transfer matrix states from artificial neural networks, Phys. Rev. B 99, 165123 (2019). A. Borin and D. A. Abanin, Approximating power of machinelearning ansatz for quantum many-body states, Phys. Rev. B 101, 195141 (2020). C.-Y. Park and M. J. Kastoryano, Are neural quantum states good at solving non-stoquastic spin Hamiltonians? Phys. Rev. B 106, 134437 (2022) Or Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neural quantum states, Phys. Rev. B 106, 205136 (2022) Yet, if we understand the problem correctly, it is most probably the other way around. Let us cite the work of Sharirr at all. [PRB 106, 205136 (2022)] here: “ By directly constructing neural-network layers that perform tensor contractions, we show that efficiently contractible TNS can be constructed in terms of polynomially sized neural networks. Our result, in conjunction with previously established results on the entanglement capacity of NQS, then demonstrates that NQS constitute a very flexible classical representation of quantum states and that TNS commonly used in variational applications are strictly a subset of NQS.”

However, this does not mean that NQS are simply better, especially not when one focuses on relatively simple ones. Tensor networks states are a well-established general-purpose ansatz for quantum spin systems. The community has been using them for decades and, therefore, these techniques are already tuned to give the optimal results. On the other hand, NQSs are still a new approach. We are still searching for the optimal strategies as well as ideal networks. But there is potential. In this respect, one should also interpret the now added direct comparison of our results with the (i)DMRG results in the expected plaquette state phase. When focusing on this problematic region, the variational energy of the RBM is worse than the one given by ED (see N=20) or (i)DMRG (see the new results in Fig. 6). However, NQS can still give the correct order parameters, i.e., the physical structure of the ground state (new Fig. 6). Besides comparing the order parameter, we also show in Appendix D the structure of the ground-state clearly showing the plaquettes. The problem of too high variational energy, especially pronounced for large lattices, is related more to the learning process than to the expected limitations of small RBM NQS themselves.

Note that comparison in the DS is not necessary as RBM NQS reproduces the exact result as we discuss in the first paragraph of the added appendix D.

The work is nevertheless a nice summary and comparative investigation of NQS, which should be suitable for publication in SciPost Physics Core after some minor revisions. I thus recommend that the authors submit a suitably revised version to SciPost Physics Core. Requested changes 1- Add some comments on the comparison to tensor network methods, specifically iPEPS.

As discussed above, we have added several comparisons of the RBM NQS results with (i)DMRG. In particular, please see Fig. 6 and the detailed discussion related to it. Some comparison (not discussed above) is also shown in Fig. 5b, however, there we compare data for infinite cylinder lattices to small periodic ones. Therefore, this example is not authoritative for the overall assessment of the NQS results. We have also expanded the paper by additional appendix D where we demonstrate that RBM is expressive enough to encompass both dimer and plaquette ordering.

2- Complete the list of references, in particular add PNAS 109, 12404 (2012). 3- Make sure that all references have a DOI (concerns in particular Refs. [4,6,7,10,11,13,27,63,64]). 4- Fix formatting of titles. Ref. [28] presents one possibility to corrupt "SrCu2 (BO3)2", Refs. [45-47] other versions, Refs. [39,40] corrupt "TmB4 ", Ref. [16] has spurious lower-casing of "Gutzwiller", etc. I note that, e.g., Ref. [18] demonstrates that the authors know how to properly format references.

We have put a significant effort into adding missing references and correcting wrong formats as well as spurious lower-casings. In addition, we have provided a thorough proofreading, corrected some typos, grammar mistakes, inconsistencies in referencing equations and figures and reformulated some sentences to make the text clearer.

We again thank the referee for the overall kind assessment of our work and recommendation for publication in SciPost Physics Core. Nevertheless, to answer the criticism of the referee, we have now significantly improved the paper. We have also added benchmarks, new strategies and new results. Therefore, and with respect to the recommendation of the two other referees, we kindly ask the referee to reconsider again the suitability of our work for the SciPost Physics.

First of all, we would like to thank the referee for the positive report as well as for pointing out some errors, missing references and for suggestions on how to improve our work. In the following, we answer in detail all the raised questions and criticism and discuss how we have improved the manuscript by reflecting on them. (Original text of the referee on which we are reacting is repeated for convenience.)

>*The results given in the paper show some potential of the method, but also many questions regarding its applicability remain open, and it is not clear how the presented results outperform already known methods. I guess, the authors need to modify the conclusions in order to uncover the power of the method for the study of the diverse properties of the Shastry-Sutherland model
>…
>4. The authors need to modify the conclusion in order to uncover the power of the method for the study of the diverse properties of the Shastry-Sutherland model.*

We have now added several direct comparisons of the NQS approach with DMRG and infinite DMRG (iDMRG) results. In particular, we have extended our analysis to larger lattices and results on lattices with mixed boundary conditions. We have focused on the regime where the ground state is expected to form the plaquette ordering. This refers to the region which is problematic already for small lattices. The reason is, that with improved learning we have been able to reproduce the exact results for the dimer phase even for large lattices. Which is now shown in our updated paper.

The comparison with DMRG in the plaquette phase showed that the finite-size scaling of the variational energy of RBM NQS for lattices with periodic boundary conditions is in a good agreement with that of DMRG. However, the learning for mixed boundary conditions proved to be much more challenging. The variational energy for large lattices (N>100) is higher than the DMRG result by several percent. Nevertheless, we were able to show that already a RBM NQS with alpha=2 is sufficiently expressive to describe the plaquette states. We now clearly state both the advantages, e.g., the fact that a very simple model without the usage of any symmetries can describe all main phases, and disadvantages, e.g. higher variational energy than predicted by (i)DMRG in the PS phase, of the approach in the conclusions.

>*1. In the introduction, the authors review the Shastry-Sutherland model. I think it is not complete, and some other references can be added, e.g.
>
>A. Abendschein and S. Capponi, Phys. Rev. Lett. 101, 227201 (2008),
>where the exact diagonalization were performed for N=36 spins;
>
>J. Dorier, K. P. Schmidt and F. Mila, Phys. Rev. Lett. 101, 250402 (2008)
>M. Nemec, G. R. Foltin and K. P. Schmidt, Phys. Rev. Lett. 101, 250402 (2008),
>G. R. Foltin, S. R. Manmana and K. P. Schmidt, Phys. Rev. B 90, 104404 (2014),
>where the model has been studied by means of the perturbative continuous unitary transformations, i.e. perturbation >theory.*


We thank the referee for turning our attention to additional literature. We have added all references to the relevant sentences in the introductory part.


>*2. The definition of the PS order parameter given in Eq. (15) is not completely correct. I mean that it is barely possible to distinguish between the "singlet" and "empty" plaquettes used in the summation. Moreover, outside the PS phase this definition has no sense. I think that the authors need to reformulate it.*


We thank the referee for pointing this out. We agree that the definition was incorrect and vague. We have corrected the formula in the updated version of the manuscript. Note that the new definition is written in a form that can be used also for uneven (Lx \neq Ly) lattices with mixed (periodic and open) boundary conditions.


>*3. In Fig. 5(c) the AF order parameter is given. However, this result is not discussed in the text. Is this result in accordance with the previous studies?*

The result is in qualitative agreement with previous works for example PRB 105(6), L060409 (2022). In the updated version of the manuscript we stress this by the following text:
Fig. 5 illustrates the usability of RBM for larger clusters. The presented results support the overall picture of the DS and AF phase separated by a narrow PS or at least its indication. Nevertheless, a much more thorough finite-size analysis would be necessary to assess the phase boundaries. For example, P_{AF} decreases with increasing system size in the whole relevant range of J′ which is in agreement with previous studies, e.g. [38]. Consequently, a careful and precise extrapolation of P_{AF} to the thermodynamic limit is needed to identify J above which the AF ordering prevails.

Please correct some typos
There are misprints in the abbreviation:
RMB .. RBM in pages 9, 13
DF .. DS in page 10
if if is suggested .. if it is suggested in page 17


We thank the referee for careful reading and for pointing out the typos. We have corrected all of them. In addition, we did a thorough proofreading of our manuscript and corrected several other mistakes, typos, grammar errors as well as unclear statements. Overall, the consideration and reflection on the questions and comments of the referee led to a significant improvement of our manuscript.

First of all we would like to thank the referee for the overall positive report as well as for the suggestions on how to improve our work. In the following we answer in detail all the raised questions and criticism and discuss how we have improved the manuscript by reflecting on them. (Original text of the referee to which we are reacting is repeated here for convenience and marked by /*** … ***/ )

/***
However, there is a very basic observation:
In the benchmark in Table 1 and in the investigation of magnetization plateaus the authors choose the values J/J'=0.2 and J/J' = 0.45, respectively, which is inside of the DS phase (at h = 0). It should be pointed out that this parameter point is not only in the DS phase but that for J/J' < 0.5 the exact ground state is given by a product state of dimer singlets, according to Shastry and Sutherland's famous solution. If I am not mistaken, this should also apply to the N=16 and N=20 clusters in Table 1. Thus at J/J' = 0.2 some of the RBM architectures even struggle to represent an exact product state of dimers. Then the question boils down to how a product state of dimers is to be represented by an RBM.
…
[1], the question of how to encode a dimer product state exactly as an RBM should be addressed
***/

The exact DS phase is always an eigenstate of the SSM and various theoretical studies suggest that it is the exact ground state up to J/J’=0.675(2) [for example, see PRX 9, 041037 (2019) or PRB 87, 115144 (2013)]. We consider this to be an exceptional quality of the model with respect to benchmarking new methods, such as NQS. It is straightforward to show that a complex-valued RBM with free biases on the input layer, i.e., the model used in the second part of our paper, is in principle capable of capturing such a state exactly. One possibility to represent a DS is to set the visible biases to reproduce the correct sign structure as discussed in the last paragraph of Sec. 3.2 and to set all the weights either to zero or a nonzero constant value. An example of such construction is now discussed in the new appendix D of the updated version of our paper. It shows that the RBM is in theory able to represent the DS state exactly. Learning the DS state is, of course, a different question.

Nevertheless, our benchmarks show that the DS can be learned [See Figs 4.(c), 5.(d) and 6(c)] even for L=100 (note that we have updated the data here) by using the second protocol. For small lattices even the first one is sufficient. We think that what is confusing here are these small but finite numbers in Table 1. They are not the lowest variation energies obtained. They are averages of the last 50 instances of the best runs. Because there are fluctuations in the variational energy due to the finite learning rate in the stochastic gradient descent as well as due to the Monte Carlo itself, these values are larger than the minimal one. We have opted for this, to have a protocol less sensitive to inaccuracies of the stochastic gradient descent method and, therefore, more reliable for the comparison of different architectures. However, the table does not necessarily show some hard limitations of the particular architecture. We have added the following note to the text, to clarify this to the readers: “Note that this means that we are not using just the lowest obtained energies but also test stability of the learning method. Consequently, the value of E50 is typically greater than zero even when the network is able to reproduce the state exactly.”

/***
[2], the benchmark of different RBM architectures should be shown also in the dimer singlet phase for a parameter point 0.5 < J/J' < 0.68, i.e. where the ground state is not an exact product state of dimer singlets.
***/

Previous studies indicate that, at least for here used finite lattices, the DS should be the ground state up to J/J’=0.675(2). Nevertheless, as also pointed out by another referee, the parameter range proposed by the referee is relevant because for example the J/J’ for SrCu2(BO3)2 at normal pressure is estimated to be 0.63. We have, therefore, added one column to Table 1, calculated for this value using the direct basis. These results are, with few exceptions, in a good qualitative agreement with the results of J/J’=0.2.

/***
Finally, the NQS fail to quantitatively describe the order parameter in the plaquette singlet phase, which is two-fold degenerate. Did the authors try to remove the degeneracy by suitably chosen boundary conditions to alleviate this problem ?
***/

We thank the referee for this question in particular. It is the main reason for our late reply as we have decided to investigate this problem in more detail. In addition, we pushed for significantly larger lattices. We have focused on the problematic region and found several interesting results. The most relevant ones we have included in the revised manuscript. First of all, already a RBM with alpha=2 is able to capture a fully developed plaquette singlet state. We tested it by using the Shastry-Sutherland lattice (graph) but setting all interactions to zero except the ones in squares where plaquettes are expected, i.e., we constructed a lattice of interacting spins on, otherwise, independent squares. RBM was able to correctly capture the expected plaquette states on all accessible lattices for both periodic (we artificially broke the degeneracy) and mixed boundary conditions. We then used these plaquette states as the initial state for the full model in the regime where the plaquette state is expected. Interestingly, for the periodic boundary conditions, this did not lead to an improvement. However, for the mixed ones it helped to lower the variational energy in the parameter regime where a significant plaquette order parameter is expected.

The introduction of mixed boundary conditions also allowed us to directly compare our results with DMRG results from Ref. [PRB 105, L060409 (2022)]. Admittedly, in the PS the RBM variational energy is typically larger than the one obtained by DMRG. Despite that, the obtained plaquette order parameters are in reasonable agreement. This again underscores the main message that just a simple NQS can capture qualitatively different orderings. The usage of mixed boundary conditions and larger lattices also lowered the estimated position of the phase transition point between the dimer state and the plaquette state to J/J’~0.68 and therefore much closer to the expected value of 0.675(2). The above points are now discussed in more detail in the main text of the manuscript as well as in the new appendix D. We have also added two multipanel figures [Fig. 6 and Fig. 10].

/***
The presentation of Fig. 6 could be improved by labeling the magnetization plateaus by their magnetization values.
***/

We have added the magnetization values to the main plateaus in the new figures 7 and 8 and also added the missing x axis label in the figure 8.

We have also focused on a thorough proofreading of our paper and corrected several other mistakes, typos, grammar mistakes as well as unclear statements. We think that all the corrections and the added results motivated by the questions of the referee led to a significant improvement of our manuscript.