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

The paper deals with frustration-free spin chains, that is, systems where the ground state minimizes the energy of each local term in the Hamiltonian separately. It is shown that the "Witten conjugation argument" from supersymmetric systems can be applied to spin chains, and used to relate many known frustration free models. In addition, it is also shown that this can be used to construct new models. Finally, these arguments are used to compute certain correlation functions (those which are distance-independent), as well as prove gaps using a "generalization of the Knabe bound" (see below) for certain parameter regimes for some of these models.

Let me say that I have enjoyed reading the paper. On the one hand, this is due to the fact that the paper is nicely written, the results are well presented, and the results are nice. On the other hand, while reading I increasingly got the impression that to a significant extent, my enjoyment was the proverbial "joy people derive from hearing something they already know" (a quote suprisingly equivocally attributed to Fermi).

What do I mean by that? Let me first try to explain what is discussed in the paper from my perspective, and then try to put the results in the paper, their relevance, etc., into context. (My perspective is by no means the only possible one, the one taken in the paper is just as valid - I will get back to that later.)

From my point, the results all fit very nicely into the language of matrix product states (MPS) and their parent Hamiltonians, and known results and techniques in that field (at least to people interested in analytical aspects of MPS).

• The Witten conjucation method is equally known in the field of MPS, where it is common knownledge without clear attribution (as far as I am aware), see e.g. https://arxiv.org/abs/1010.3732 or https://arxiv.org/abs/1403.2402). Specifically, a parent Hamiltonian h>=0 can be deformed by multiplying it with M_h^\daggerhM_h, while the ground state is deformed in the converse way, and where M_h is a product of the local deformations acting on the support of h; in addition, since only the kernel of h matters, it can be changed at will on its support. This corresponds exactly to the transformations M and C (changing h on the support) mentioned in the paper. This reasoning is independent of having an MPS ground state/space, as also follows from Witten's argument.

• Since in this procedure, the ground state/space is changed in the converse way (i.e. by multiplying with M^(-1), which is a product of local operations), it follows immediately that the ground space of the deformed model is still parametrized by MPS.

• For instance, an argument akin of this is already given in Refs. [34] and [40] for q-deformed AKLT models (dating back to 1991/1992), where a formula very similar to Eqs. (1-3) of the paper is given (Eq. 7 in [34], eq. 8 in [40]), which also discusses their MPS form.

• If the MPS in question have a nice form - e.g., they are product states, or GHZ-type states (i.e. the Ising or Potts states from the paper, e.g. Eq. (65)), then a wide range of known results and techniques on MPS and their ground state parametrization apply. In particular, the ground state will always be long-range ordered, so using the suitable canonical forms one can determine the symmetry broken states, which again have a nice canonical form. In particular, this implies that in the theromdynamic limit, the correct physics is captured by the symmetry broken states, and no superpositions need to be considered for correlation functions etc. (nor should they be considered, as done in the paper).

• This property holds for all examples except for the one derived from the Heisenberg ferromagnet (Sec. 3.3).

• Therefore, in all those cases, the exact correlation functions for any observables can be computed analytically (or almost analytically, depending on the form of the transfer matrix) using well-established MPS techniques.

• Finally, there are powerful tools for proving gaps for parent Hamiltonians of MPS, both with unique and with finitely degenerate ground space (again except for the Heisenberg ferromagnet) (see Nachtergaele, Commun. Math. Phys. 175, 565 (1996) for the more general degenerate case). Thus, unless the Hamiltonians constructed for some reason are completely unrelated to parent Hamiltonians (which is very unlikely, if at all possible; especially in the cases like the Ising or Potts model where they have been obtained by deforming parent Hamiltonians, this will definitely not be the case), these results imply that the models are gapped for the full parameter regime.

• One point one might find surprising about the results presented is that all these models have exact MPS ground states (even if it is not stated like that in the paper - but as argued, locally conjugating MPS, such as the Potts ground state, again give MPS). But in fact, this is not too surprising: A result of Matsui [Infinite Dimensional Analysis, Quantum Probability and Related Topics 1, 647 (1998)], later refined by Ogata, shows that any frustration free gapped Hamiltonian with a bounded ground space degeneracy in the finite volume will have a ground space parametrized by MPS. Thus, as soon as we have sufficient evidence that the Hamiltonian satisfies these properties, it should have an MPS ground space, and the task is only to find it. And then, one can use the full power of the MPS machinery to assess the properties of such frustration free models, which allow for the strong statements given above.

So what does all this this mean with regard to the current paper?

Of course, the fact that these results can be derived in the language of MPS don't invalidate or devaluate the results presented - even if I feel that even in that case, including this perspective might be elucidating.

However, my concern is that the MPS perspective on the one hand supersedes some claims of novelty, and on the other hand allows to prove a significantly stronger version of some of the claims made in the paper.

Specifically, it is repeatedly claimed (e.g. in the introduction and conclusion) that this technique is new, that the paper transfers Witten's technique to spin chains, etc., which I think is not a valid claim in the light of the fact that this is a well-known technique in the field of MPS.

At the same time, some new findings of the paper are the computations of certain correlations and a gap proof. However, both of these could be significantly strengthened using the techniques mentioned above.

In particular, the correlation functions discussed in the paper are special, distance-independent correlation functions which are moreover computed in specific, not symmetry broken ground state: First of all, one could thus argue that one should instead consider the correlations in the symmetry broken state, but, much more importantly, one can easily obtain all correlation functions from the MPS description (both of the symmetric and the symmetry broken states), so there is not need to restrict on those very special ones.

Regarding the gap, similarly much stronger statements should be possible if one uses the results of Nachtergaele quoted above, which should allow to prove a gap for the whole range of parameters. Again, the application of these techniques is quite straightforward. (The Knabe-type argument used in the paper is these days rather used in 2D etc., where no general argument similar to Nachtergaele exists.)

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.

Finally, some smaller comments:

Regarding the "generalisation of the Knabe bound", "generalisation" seems a bit overstating the result/technique. It is simply an application of a basic technique (H^2>cH, expanded in terms, and dropping certain terms, leading to a local bound as in Corollary 2) to the given scenario, which has been done in quite a few contexts and is quite standard to those working on this topic. It certainly does not generalize the Knabe bound in any strong sense. (There is also some debate whether one should call this a "Knabe bound", since the latter often refers to relating the spectrum of the same* Hamiltonian on OBC and PBC (such as a 1/N gap on a finite ring in Knabe's original work, or a 1/n^(3/2) gap for OBC in 2D in [60]) , and also since the basic technique used here is equally used in the martingale method. But I don't think there is a full agreement how to call the inequality of Corollary 2, so "Knabe method" is certainly ok.)

Doesn't the inequality in (70) simply follow from the fact that the spectrum of simga_j is <=1, independent of Cauchy-Schwarz?

Dear Editor,

Frustration-free models with solvable ground states are one of the best tools we have in understanding the emergent behavior of many-body quantum systems. Accordingly, a multitude of such models have been uncovered and studied over the years. The present work unifies many of these models by showing how they can be derived from a new unifying principle, and also sheds new light on recent works on Z3 parafermion chains. Moreover, the authors use this novel method to derive a whole range of new frustration-free models, exemplifying the applicability and generality of their method.

This paper is clearly written and the arguments and derivations are convincing, as are the demonstrations of deriving known and new models. This is clearly a powerful method that can be used to efficiently construct new frustration-free models. Moreover, although the examples discussed in this work are 1D and nearest-neighbor, it is clear that the method itself can be applied to higher dimensions and longer-range systems without needing modification.

While this is a very nice work, I can not yet wholeheartedly recommend publication of the article in its present for since there are a few points that can be improved upon: 1) Firstly, the authors credit their idea to Witten. It would be instructive to give the reader some sense of how similar/different the authors' Theorem 1 is to what Witten did. 2) This new method is really impressive due to its generality. The authors are diligent in citing a lot of previous works on specific frustration-free models. However, I do not see any discussion of previously known principles of generating frustration-free models. E.g., one such method is the parent Hamiltonian construction of MPS ground states. Perhaps/presumably there are other methods as well? Have the authors made sure that they did not overlook any known methods? This is of course a very important point, and it would help if the authors discussed previously-known principles for constructing frustration-free models, such that the reader can accurately place this new development in its proper context.

Less crucial, I have a few minor questions/suggestions: * 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. * 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? * It might be useful to plot the frustration-free lines in some phase diagrams, to see what they look like in parameter space. * 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? * 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?