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^\dagger*h*M_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>c*H, 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?

1-generality of method
2-unifying previous models
3-construction of new frustration-free models

1-lacking a careful discussion of previously-known methods

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?