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Constructive approach for frustration-free models using Witten's conjugation

by Jurriaan Wouters, Hosho Katsura, Dirk Schuricht

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Authors (as registered SciPost users): Hosho Katsura · Dirk Schuricht · Jurriaan Wouters
Submission information
Preprint Link:  (pdf)
Date submitted: 2020-11-02 10:01
Submitted by: Schuricht, Dirk
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We extend Witten's conjugation argument [Nucl. Phys. B 202, 253 (1982)] to spin chains, where it allows us to derive frustration-free systems and their exact ground states from known results. We particularly focus on $\mathbb{Z}_p$-symmetric models, with the Kitaev and Peschel--Emery line of the axial next-nearest neighbour Ising (ANNNI) chain being the simplest examples. The approach allows us to treat two $\mathbb{Z}_3$-invariant frustration-free parafermion chains, recently derived by Iemini et al. [Phys. Rev. Lett. 118, 170402 (2017)] and Mahyaeh and Ardonne [Phys. Rev. B 98, 245104 (2018)], respectively, in a unified framework. We derive several other frustration-free models and their exact ground states, including $\mathbb{Z}_4$- and $\mathbb{Z}_6$-symmetric generalisations of the frustration-free ANNNI chain.

Author comments upon resubmission

——————————————— Reply to referee 1 ——————————————— We thank the referee for his/her very positive report and useful remarks to improve our manuscript. In order to make the relation to Witten’s original conjugation argument more transparent we have added an appendix to discuss this in detail. To address the 2nd point we have extended the introduction to also include some discussion of parent Hamiltonians and MPS.

Below are our answers to the minor points raised by the referee: * In Eq.(15), what gives the condition on r? E.g., why can we not make r negative, or complex? Naively, if r=-1, then the operator M does not do anything, but the Hamiltonian (Eq.(20)) is now mapped to the YY chain, so I guess somewhere the condition r>0 was explicitly used? It is not clear where.

We considered only positive r to keep the system simple, but indeed other choices for r are possible.

  • If in Eq.(21) we took the r parameter to be independent of the r parameter in (15), do we get a two-parameter family of frustration-free Hamiltonians? Is there a reason to take them to be the same r?

In principle the parameters in (15) and (21) can be chosen independently. However, this will lead to quite complicated expressions for the deformed Hamiltonian. In contrast, our choice leads to the well-known ANNNI model which we included also as reference to later discussions (see Sec. 6).

  • It might be useful to plot the frustration-free lines in some phase diagrams, to see what they look like in parameter space.

In general it is hard to include the frustration-free lines into phase diagrams since, apart from the simple cases like the ANNNI model (Sec. 3.2), the Hamiltonian will possess too many terms to allow a nice presentation in a two-parameter setup. We realised, however, that a more schematic presentation of the link between the models discussed in Refs. 2 and 3, which constitutes one of our main results, can be included, which we now do as new Fig. 3.

  • I agree with the authors that their method is application to further-range and higher-dimensional models. Have the authors thought of perhaps including an example just to make this point?

We are currently working on several two-dimensional frustration-free models. However, as including such models would require the introduction of additional notation, we decided not to include them in the present manuscript.

  • Lastly, this is more for my own curiosity, since I can imagine that it is hard to prove one way or another: do the authors expect that all known frustration-free models in 1D can be understood in this way?

While it is tempting to believe that such a universality of the approach is indeed the case, we certainly would not like to make the claim that no exceptional, frustration-free models can be found that evade the conjugation construction.

——————————————— Reply to referee 2 ——————————————— We thank the referee for his/her extensive report. The referee is absolutely right that many of our results can be rephrased in terms of matrix product states, which may provide a more general setting. We are glad that the referee acknowledges that this is, however, not the only point of view that can be taken. In fact, we decided to keep the presentation of our results in its present form, since we believe that this allows a less abstract discussion which is more suitable for a general audience. In order to make the link to matrix product states we have revised the introduction accordingly.

As pointed out by the referee, the formulation in terms of matrix product states allows the application of tools based on parent Hamiltonians to prove the existence of finite energy gaps. Including this in our manuscript, however, would have required a more abstract reformulation of our results, which we refrained from doing as discussed above. Still, we decided to improve our analysis of the energy gaps by considering longer sub-chains in the Knabe method (see revised App. B), which allowed us to extend the bounds on the gapped regions while keeping the presentation at a less abstract level.

Similarly, it is possible to extend the results on the correlation functions either by direct calculations or using matrix product state methods. However, in our manuscript we only give the simplest example of a correlation function (which is indeed independent of distance) to point out how correlation functions can in principle be obtained in the considered systems. We have clarified the text around (70) accordingly.

One specific remark we would like to answer concerns AKLT models: The equations in the papers pointed out by the referee mean that positive semidefinite local Hamiltonians can be written in the form of A^\dagger A. This universally holds in any frustration-free model (provided that the ground state energy is set to zero.) But this form does not immediately imply the connection between q-deformed and undeformed AKLT models, which we discuss in this manuscript. 

We hope that the referee is content with our decision to keep the presentation less abstract, ie, not to rephrase our results in terms of matrix product states. We have aimed to include some discussion of these more recent developments in the introduction.

List of changes

-revised introduction to discuss parent Hamiltonians and matrix product states, and add a remark clarifying their relation to our work
-clarified the discussion of correlation functions around (70)
-added Fig. 3 in the conclusion to illustrate our result linking the two frustration-free models of Refs. 2 and 3 to the classical Potts model
-added App. A to clarify the differences between Witten’s construction in the supersymmetric case and our version for spin chains
-revised App. B to improve the bounds obtained by Knabe’s method, updated the resulting bounds given in the main text
-corrected a few typos and updated some references

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-3-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2005.12825v2, delivered 2021-03-22, doi: 10.21468/SciPost.Report.2727


After the reply by the authors and the revision of the manuscript, my key
points of criticism still remain unchanged. In particular, the last
paragraphs of my report apply unchanged, and I therefore append them at
the end of the report. Let me just stress a few points:

The key issue I see (see the original report below) is that (1) the
techniques are not new and (2) the paper is mostly a collection of models
which are analyzed to some extent using the technique. Each of these
analyses is technically not challenging and more on the level of an
exercise problem. The value of this paper would therefore stem from its
comprehensiveness, which I don't seen as given, given that it chooses to
completely ignore the powerful yet simple-to-use tools derived in the last
almost 30 years.

I have to say that I find the way in which the existence of the MPS
results has been included in the introduction rather disappointing; this
will serve to perpetuate the misbelief that proving gaps of frustration
free 1D systems is rather hard and that therefore, for many models such
gaps are not known. It also gives the impression that the gap results by
Fannes-Nachtergaele-Werner and Nachtergaele are hard to apply, which is
not the case - they state rather clear theorems which don't require a
particularly high degree of technicality to apply them. In particular,
Theorem 2 in Nachtergaele shows immediately the *existence* of a gap,
without any further ado, and also the bounds on the gap are quite
straightforward to apply and do not require refined tools (not to prove
the results! - but that's why one can rely on previous work, to not have
to re-prove things oneself). Let me add that I find it rather disturbing
that a paper classified as "Mathematical Physics", both on SciPost and on
the arXiv, would use the need for a "more abstract reformulation" as an
argument against using these bounds (especially given what "abstract"
means in this specific case).

Regarding the correlation function, let me iterate again that I don't
really see the value of computing the distant-independent correlation
function in a cat state: These states are completely unphysical, and the
symmetry broken states have simply no correlations. Generally, physical
states with distant-independent correlation functions would have no
correlations (or otherwise be unstable). Again, using MPS would give
relevant results away from zero correlation length, without introducing
any particularly technical aspects beyond MPS as such.

So overall, I don't see sufficient grounds for this paper to be suitable
for SciPost, but I also feel that it is overall an editorial decision
whether the type of material in the current form is suitable for SciPost.
For completeness, let me attach the key points from my last Report which I
feel still fully apply and capture the key point of my scepticism, and
which I would like to strongly re-emphasize:


So my impression is that from the point of view of techniques, the
techniques presented in the paper are not new, and in terms of results,
that one should be able to get significantly stronger results with
state-of-the-art techniques.

So what remains? As far as I can tell, the paper treats a wide range of
models in a unified language, which can be understood in terms of MPS.
Each of these treatments by itself is rather simple and more on the level
of an exercise problems. The value of the paper thus stems from the
amount of the results presented, which is indeed quite impressive (though
there are certainly even more models out there which fit in the
framework), and the comprehensiveness of the treatment (which as of now is
not given, as discussed above).

So is a collection of models, where it is shown that they all fit in the
same framework, sufficient to merit publication in SciPost? I don't know,
and I think it is more of an editorial decision; all I can provide is my
assessment of the results. My feeling is that in some sense, this has
more of a review-type character, but this is not necessarily a bad thing.

However, one thing I feel is that given the type of results presented -
i.e., showing that a wide range of models can be analyzed using the same
language - the paper should do so *comprehensively*. To me, this means
that it should not stop in 1990, but take the more contemporary
MPS/finitely correlated states perspective into account, and derive the
strongest results possibe, that is, find the MPS representation of the
ground states, possibly the behavior of relevant correlation functions -
and otherwise make clear they can be easily computed from the MPS
description - , and prove the gap for the full range (and if for some
reason these general techniques don't apply, make clear why they don't).
(Indeed, the central results in that regard - Fannes/Nachtergaele/Werner
and Nachtergaele are from the early 90ies, and also Matsui's result is
from the 90ies, so it not even that "contemporary".) I think this is even
more so important in order to avoid and not perpetuate the not so uncommon
misbelief that proving gaps for valence-bond-type wavefunctions (i.e. MPS)
in 1D is a hard problem, as witnessed by the fact that this was hard work
in the original AKLT paper - while this is of course true, the quoted
followup works have established general tools to assess these problems,
setting the problem at rest, and these results should not be ignored.

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Anonymous Report 1 on 2021-3-8 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2005.12825v2, delivered 2021-03-08, doi: 10.21468/SciPost.Report.2660


In my first report, I was very positive about the nice method introduced in this paper: using neatly worked-out examples, this paper shows how to construct paths of frustration-free Hamiltonians. The main concern I raised in my report was that the authors lacked a discussion of/comparison to existing methods: in particular, it was not clear whether the authors had done their due diligence to ensure that their method is, in fact, novel as claimed.

The second referee has pointed out that the conjugation method is in fact already well-known in the MPS literature (backed up by concrete references listed in the referee's report). This is a serious issue, which the authors do not seem to address in their reply to that referee. They mainly focused their reply on their decision to not rephrase their results in MPS notation. I have no objection to that choice, but that does not lift them from their responsibility to correctly credit previous works. Even in version 2, the authors clearly claim that they are introducing a new method. Either the authors disagree with the second referee's claim that this method is well-known (in which case they should explicitly address this issue in their reply), or the authors agree and should clearly weaken their claim of novelty.

While I hope to have made my point about the importance of the above issue, let me also say that even if the conjugation method is not new, the present work is still a valuable addition to the literature since it makes this method more readily accessible to a wider audience. As such, I can still see it having a place in scipost, but based on the merit of elucidating and explaining a known method (instead of introducing a fundamentally new method); this is a subtle issue where I cannot give a clear-cut recommendation to the editor. Either way, publication of this paper should not proceed before the authors address the issue of the claimed novelty of their method, which seems to be in contrast to the second referee's claims.

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