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Permanent variational wave functions for bosons

by J. M. Zhang, H. F. Song, Y. Liu

Submission summary

Authors (as registered SciPost users): Yu Liu · Jiang-min Zhang
Submission information
Preprint Link:  (pdf)
Date submitted: 2021-11-19 07:18
Submitted by: Zhang, Jiang-min
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational


We study the performance of permanent states (the bosonic counterpart of the Slater determinant state) as approximating functions for bosons, with the intention to develop variational methods based upon them. For a system of $N$ identical bosons, a permanent state is constructed by taking a set of $N$ arbitrary (not necessarily orthonormal) single-particle orbitals, forming their product and then symmetrizing it. It is found that for the one-dimensional Bose-Hubbard model with the periodic boundary condition and at unit filling, the exact ground state can be very well approximated by a permanent state, in that the permanent state has high overlap (at least 0.96 even for 12 particles and 12 sites) with the exact ground state and can reproduce both the ground state energy and the single-particle correlators to high precision. For a generic model, we have devised a greedy algorithm to find the optimal set of single-particle orbitals to minimize the variational energy or maximize the overlap with a target state. It turns out that quite often the ground state of a bosonic system can be well approximated by a permanent state by all the criterions of energy, overlap, and correlation functions. And even if the error is apparent, it can often be remedied by including more configurations, i.e., by allowing the variational wave function to be a combination of multiple permanent states. The algorithm is used to study the stability of a two-particle system, with great success. All these suggest that permanent states are very effective as variational wave functions for bosonic systems, and hence deserve further studies.

Author comments upon resubmission

We have added a subsection to illustrate the application of the algorithm to a two-particle stability problem.

It is a cubic lattice as large as 21*21*21.

The overlap is amazingly high.

List of changes

(1) The first and last sentences of the abstract are rewritten.

(2) The fourth paragraph of the Introduction is greatly expanded.

(3) A few sentences are added in the last paragraph of the Introduction. They are

'Note that this was not considered previously, but is necessary and effective for improving accuracy.'

'This enables us to use the algorithm to study the stability of a two-boson system, which is analogous to the negative ion of hydrogen.'

'We would like to mention that the whole paper is actually a by-product of studying these open problems.'

(4) The subsection containing equation 74 is new. It deals with a two-particle stability problem in a 3d lattice.

(5) The paragraph containing equations 56 and 57 is new.

(6) In the second paragraph of the Conclusion, we have added 'Note also that in practice many models of interest have only $N=2$ or $N=3$ particles. For such small values of $N$, the permanent computation is of course not an issue at all.'

(7) The 3rd paragraph of the Conclusion is new.

(8) The 4th paragraph of the Conclusion is expanded.

(9) In the 3rd last paragraph of the Conclusion, the following sentence is new: 'Given a primitive wave packet $\phi(\vec{x})$, we can imagine a many-body permanent state constructed with the wave packets $ \phi(\vec{x} - \vec{R}_m)$, where $\vec{R}_m $ runs through an $n$-dimensional lattice.'

(10) In the bibliography, refs. 41-44, 46, 55 are new.

Current status:
Publication decision taken: reject

Editorial decision: For Journal SciPost Physics: Reject
(status: Editorial decision fixed and (if required) accepted by authors)

Reports on this Submission

Anonymous Report 4 on 2021-12-17 (Invited Report)


I've read through the correspondence and I tend to agree with the other reviewers in their assessment.

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

Anonymous Report 2 on 2021-12-17 (Invited Report)


I have read the paper and the past, rather long, correspondence about it.
I agree with the general opinion of the two referees that, although this is a detailed work on how to define and to implement the permanent trial wave function for a bosonic system, it does not meet the criteria for acceptance. Its main limitation is the missing of a clear application to a previously unsolvable problem.

  • validity: good
  • significance: low
  • originality: ok
  • clarity: ok
  • formatting: good
  • grammar: good

Anonymous Report 3 on 2021-12-13 (Invited Report)


1. The paper is well written and easy to follow.
2. The authors went into enormous depth in exploring this variational method.


1. The main weakness is that it is not obvious to me that this method has any relevant future application
2. I am missing a comparison with other variational wavefunctions for bosonic systems


I was asked to review this paper after the paper had been subject to two rounds of refereeing. I have read the issues raised by the other reviewers as well as the authors' reply.
In general I am on the same page as the other Reviewers. The proposed method is analyzed in great depth, supported by mathematical theorems, and implemented with state of the art algorithms. It is properly benchmarked for the 1D Bose-Hubbard model up to 12 particles. Different classes of variational wavefunctions are provided and compared. Optimization strategies are discussed. Special emphasis is on the possible non-orthogonality of the variational wavefunctions. The provided examples are for the Bose-Hubbard model in 1D and other few-boson problems. All of this is proof of a careful and complete analysis.
But the main problem with the method is that I don't see a path to a future relevant application, and with this I am on the same page as the other reviewers.
I would like to add one more point however: What I am missing is a comparison to other variational approaches. For bosons, the best known variational approach is the Gutzwiller wavefunction (D.S. Rokhsar and B.G. Kotliar, Phys. Rev. B 44, 10328 (1991), D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998)) -- and there are many extensions of the simplest scheme. Another, lesser known example is the Baeriswyl wavefunction (Phys. Rev B vol. 93 art.174518 (2016)). How does the accuracy of the permanent compare to the simplest Gutzwiller method?

Requested changes

1. The critical value of the superfluid to Mott insulator transition in 1D is about 10% smaller than what the authors write.
2. A discussion on Gutzwiller wavefunctions is requested.

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

Author:  Jiang-min Zhang  on 2021-12-18  [id 2036]

(in reply to Report 3 on 2021-12-13)
suggestion for further work

Thanks a lot for spending time on the review.

This is just a preliminary reply.

Note that in our scheme, the particle number is fixed, while in the Gutzwiller variational wave function, the particle number is fluctuating. Therefore, we do not know how to compare the two variational wave functions. This is especially the case when the particle number is as small as N=2. For such an extremely dilute system, we do not even know how to implement Gutzwiller.

We have never seen data on the overlap between a Gutzwiller wave function and the exact ground state. But we can hardly image that such a crude variational wave function has higher overlap than ours, as there is little room for improvement. Our wave function can reproduce the one-particle correlation function almost exactly, we would be shocked if the Gutzwiller can be as good or even better.

The Baeriswyl wave function is of more interest to us. At least, it is particle-number conserving. We shall look into it. Thanks a lot for pointing it out to us.

Using the algorithm to study the structure of a bosonic wave function will be our next project.

Anonymous Report 1 on 2021-12-12 (Invited Report)


1. Accurate presentation of the permanent and of its properties;
2. Clear and well-written presentation.
Strengths B [As from the revised version of the manuscript]:
3. [Additional] statement of the implementation of the method and of its applicability;
4. [Additional] examples of applicability of the method.


1. Work focused onto a calculation method: very limited applications to more-or-less scholastic problems presented as specific examples.


On my second review of this manuscript while, on one hand, I acknowledge the effort that the authors have put in answering the various referee remarks, on the other hand I must say that they are over-all only partially convincing about the publishability of the manuscript in Sci Post.
The authors definitely managed to make it clear that their use of the permanent eventually leads to a reliable variational many-particle state for bosonic systems and that the whole procedure is systematically optimized by an algorithmic choice of the best set of single-particle wavefunctions to use as the building blocks of the permanent. Thus, I now can see how the approach outlined here has to be regarded in relation to alternative numerical methods, such as the ones mentioned in my previous report.
The newly added subsection V-E is useful as a further illustration of the applicability of the model: this further enforces my previous judgement about the paper being clearly written and easy to follow. Over-all, I still keep my opinion that the topic of this manuscript is about a method, without reference to any relevant progress in a specific physical problem. I acknowledge that there are several potential further applications of the method, as outlined by the authors, but none of them is specifically mentioned here.
For this reason, while I believe that this is definitely a well-written paper that warrants publication in some form in a technical journal, I doubt that SciPost is the appropriate place where to publish it (see my points below). In its present form, I therefore do not recommend publication of the paper in SciPost. To warrant publication in SciPost, I believe the authors should largely expand the discussion about one of the points they list in their reply to the first referee’s comment to Expectation number 4. Expicitly referring to the authors' reply, in the following I separately list the comments I did on my previous report, the authors' reply, and my further comments in this second review:


1. Detail a groundbreaking theoretical/experimental/computational discovery:

[First Report] The paper clearly deals with a computation method. In order to see a specific advancement on the computational side I think a careful comparison should be made with alternative numerical methods for what concerns the typical system size one can deal with (of course for similar systems), the over-all time required to run algorithms computing the same thing in the same system with different techniques (such as, e.g., the authors' method, the DMRG approach, the Monte Carlo techniques, and so on) to assess whether the method proposed here has, at least, the same level of reliability of alternative, well-grounded and widely used approaches. In this manuscript I see nothing pointing in that direction.

[Authors' reply] Our reply: Sorry, but we cannot make such a detailed comparison for the following

1. We are not experts in Monte Carlo or DMRG. We really cannot learn them from scratch now.

2. It is apparent that the current algorithm has its advantages. DMRG works well only for 1d systems with short range hoppings and interactions. The current method works well for 3D, and the hopping and interaction can be arbitrary. As for MC, it can suffer from the negative sign problem if there is frustration, but that is not a problem for us. In many cases, it should be slower than us. Furthermore, the current algorithm can be easily modified for fermions, for which MC might meet great difficulty.

3. It is apparent from the algorithm that the memory cost is minimal. It does not increases with the particle number N (unlike exact diagonalization). In a newly added subsection, we treat a two-particle system with L = 9261 sites. The memory cost is that of a 9261*9261 matrix. It is the same for N = 10.

4. As for time, it can be seen that if the particle number N is around 10, the permanent computation is something. But it is still okay---in our paper, figure 12 (N = 11) is most time-consuming, but it is still less than 24 hours (it would be less than 8 hours if we just want fig. 12a). Most figures can be done in 30 mins with Matlab. For fixed N, if the system size L increases, solving the generalized eigenvalue problem could become the bottle neck in the end. But the complexity of solving a generalized eigenvalue problem increases as L^3 only. We have actually collected the relevant data, but we really do not want to display it because it can be inferred from the algorithm. We have actually discussed it in the conclusion part before.

5. The current algorithm is a wave-function based method. Once we have the wave function, we can calculate any quantity we like. The philosophy is totally different from those of MC or DMRG. It is unlikely that a method can be substituted by another one with a completely different philosophy.

[Second report] So, now I finally see that this is a smart implementation of a variational method. Is it really enough to warrant publication in SciPost? I believe it is not.

2. Present a breakthrough on a previously-identified and long-standing research stumbling block:

[First Report] The proposed applications of the authors' method are limited to a small number of well-known one-dimensional lattice models with a small number of sites (a lattice as large as about 10 sites) and even in that case the reliability of the results is apparently not better that what one would possibly get by using alternative numerical methods. I do not see any specific long-standing problem that might be (even partially) tackled by resorting to the technique proposed in this paper, better than with some alternative method.

[Authors' Reply] Our reply: To prove that the algorithm can work for higher dimensions, we have added a subsection (the one containing equation 74). It is a cubic lattice, as large as 21*21*21. The permanent approximation is amazingly good. The algorithm is faster than exact diagonalization (ED). It can also handle bigger lattices than ED. With the algorithm, we got know that the exact ground state can be exceedingly well approximated with a single permanent. We do not know whether MC can deliver the same insight. Moreover, once we have a simple wave function, we can calculate any physical quantity as we like, there is no need to restart the algorithm.
In this tentative work, we first need to benchmark the algorithm against ED. We can easily handle N=12 particles on a lattice of size L = 100. It is as easy as the N=12 and L = 12 case. But for such a system size, we have no ED for comparison. Note that our primary concern is whether a ground state can be well approximated with permanent states---We really care about the overlap. So, we need the exact ground state, which can only be delivered by ED.

[Second Report] I appreciate the authors’ answer. However, my point was
“I do not see any specific long-standing problem that might be (even partially) tackled by resorting to the technique proposed in this paper, better than with some alternative method.”
No specific long-standing problems are mentioned in the authors’ reply. Their method can be possibly easy to learn and to implement, but still there is no mention of an actual problem that can be efficiently solved by their method.

3. Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work:

[First Report] The only immediate possible follow-up work, along the direction of this manuscript, that I can see could be extending the system size and possibly checking the scaling of e.g. computation time, estimates of the energy and/or of the state overlap, and so on, with the system size. But, again, without a serious comparation to what one could do with alternative numerical methods I do not see why one should push forward this research line.

[Authors Reply] Our reply: To be frank, we are not interested in a detailed but laborious comparison. The current algorithm is in spirit like a Hartree-Fock method. The Hartree-Fock method is not replaced by other methods. The paper is already too long. Making it longer would only reduce the readability.
We can upload the codes to Github, so people can compare if they like.
As for possible follow-up work, please see our reply to the first referee.

[Second Report] I saw the reply. The only place where I can see a reference to opening a pathway in existing research direction is the sentence
“With the algorithm, we can numerically search for the optimal permanent approximation of a bosonic wave function, the overlap will provide a geometric measure of the entanglement in the wave function. Hence, it might be of interest for people in quantum information. MC cannot serve such purposes.”
Quantum information is a continuously developing field: many topics follow under the general cathegory of quantum information. I see no mention to any of them in which a new research direction can be opened by the authors’ method

4. Provide a novel and synergetic link between different research areas:
I cannot see immediate chances for synergetic links between different areas arising from this work.

[Authors' Reply] Our reply: Please see our reply to the first referee.

[Second Report] I saw the reply. I acknowledge that, at this point, there might be a few relevant issues in the authors’ answer. However, I believe that more work has to be done in order for this manuscript to warrant publication in SciPost.


General acceptance criteria:

1. Be written in a clear and intelligible way, free of unnecessary jargon, ambiguities and misrepresentations:

The paper does meet all the above requirements: it is well written, free of unnecessary jargon and more-or-less self-contained, for what concerns the definition and the properties of the permanent.

2. Contain a detailed abstract and introduction explaining the context of the problem and objectively summarizing the achievements:

The context of the problem is explained at a good enough level of details.

3. Provide sufficient details (inside the bulk sections or in appendices) so that arguments and derivations can be reproduced by qualified experts:

The various mathematical steps are well documented and detailed. The paper does meet this requirement.

4. Provide citations to relevant literature in a way that is as representative and complete as possible:

The paper meets this requirement.

5. Provide (directly in appendices, or via links to external repositories) all reproducibility-enabling resources: explicit details of experimental protocols, datasets and processing methods, processed data and code snippets used to produce figures, etc.:

The paper meets this requirement.

6. Contain a clear conclusion summarizing the results (with objective statements on their reach and limitations) and offering perspectives for future work:

The paper meets this requirement.

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

Author:  Jiang-min Zhang  on 2021-12-18  [id 2035]

(in reply to Report 1 on 2021-12-12)

We are happy that the referee now agrees that the algorithm is smart, easy to learn and easy to implement.

As for the criticism of lack of applications to more-or-less scholastic problems, we are now thinking of applying it to fermions. For fermions, we have determinants, which are much easier to calculate than permanents, yet the strategy is still the same.

As we can see, we have failed to convey the key point that a fermionic or bosonic wave function might have deep structures.

Anyway, we are grateful to the referee for spending time on the review.

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