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Accuracy of Restricted Boltzmann Machines for the one-dimensional $J_1-J_2$ Heisenberg model

by Luciano Loris Viteritti, Francesco Ferrari, Federico Becca

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Submission summary

Authors (as registered SciPost users): Francesco Ferrari · Luciano Loris Viteritti
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-05-02
Date submitted: 2022-04-15 16:04
Submitted by: Viteritti, Luciano Loris
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approach: Computational


Neural networks have been recently proposed as variational wave functions for quantum many-body systems [G. Carleo and M. Troyer, Science 355, 602 (2017)]. In this work, we focus on a specific architecture, known as Restricted Boltzmann Machine (RBM), and analyse its accuracy for the spin-1/2 $J_1-J_2$ antiferromagnetic Heisenberg model in one spatial dimension. The ground state of this model has a non-trivial sign structure, especially for $J_2/J_1>0.5$, forcing us to work with complex-valued RBMs. Two variational Ans\"atze are discussed: one defined through a fully complex RBM, and one in which two different real-valued networks are used to approximate modulus and phase of the wave function. In both cases, translational invariance is imposed by considering linear combinations of RBMs, giving access also to the lowest-energy excitations at fixed momentum $k$. We perform a systematic study on small clusters to evaluate the accuracy of these wave functions in comparison to exact results, providing evidence for the supremacy of the fully complex RBM. Our calculations show that this kind of Ans\"atze is very flexible and describes both gapless and gapped ground states, also capturing the incommensurate spin-spin correlations and low-energy spectrum for $J_2/J_1>0.5$. The RBM results are also compared to the ones obtained with Gutzwiller-projected fermionic states, often employed to describe quantum spin models [F. Ferrari, A. Parola, S. Sorella and F. Becca, Phys. Rev. B 97, 235103 (2018)]. Contrary to the latter class of variational states, the fully-connected structure of RBMs hampers the transferability of the wave function from small to large clusters, implying an increase of the computational cost with the system size.

Author comments upon resubmission

Dear Editors,

we would like to resubmit the attached revised version of the manuscript which includes the suggestions made by the referees. We reply to the comments of the referees highlighting the changes made to the manuscript.

Sincerely yours,
Luciano Loris Viteritti, Francesco Ferrari, Federico Becca

List of changes

List of changes:
- we corrected the typos;
- we added a paragraph in the introduction to answer the question of referee 1 about the usefulness of RBMs for 2d frustrated systems and the comparison with DMRG approaches;
- we add a comment in the results and in the caption of Fig. 6 to clarify the difference between a N_{MC} and an N_{opt} step as required by referee 2;
- we specify that the overlap in Fig. 12 between the variational and the exact states should be understood in modulus to answer the question of referee 2.

Published as SciPost Phys. 12, 166 (2022)

Reports on this Submission

Anonymous Report 2 on 2022-4-25 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2202.07576v2, delivered 2022-04-25, doi: 10.21468/SciPost.Report.4986


see previous report


see previous report


The revised version is adequate for publication in SciPost Phys.

Requested changes

1- (optional) However, I still recommend that the author make an earlier mentioning of the pBCS method (i.e., in the intro) and move the contents of the Appendix before Sec. 3. Knowing what this pBCS method is essential for the reader to be able to comprehend the comparions.

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

Anonymous Report 1 on 2022-4-24 (Invited Report)


I have looked through the author's modifications and replies.
I recommend the paper be published in SciPost.

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

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