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Interrelations among frustration-free models via Witten's conjugation
by Jurriaan Wouters, Hosho Katsura, Dirk Schuricht
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Submission summary
| Authors (as registered SciPost users): | Hosho Katsura · Dirk Schuricht · Jurriaan Wouters |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202104_00026v1 (pdf) |
| Date submitted: | 2021-04-24 08:57 |
| Submitted by: | Schuricht, Dirk |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We apply 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 positive report and useful remarks. Indeed we fully agree with the referee that one of our main contributions is to elucidate the Witten conjugation method to the condensed matter community. We then use it to provide a unified framework to discuss several recently studied frustration-free models, a motivation that is now also highlighted by the revised title. We believe that the revised manuscript is clear that we do not introduce a fundamentally new method, but rather apply Witten’s conjugation to achieve this aim. In order to clarify the relation to existing MPS works we have extended the introduction (new 6th paragraph).
Reply to referee 2
We thank the referee for his/her report. We have extended the discussion on the application of Witten conjugation in the MPS framework and added further references. We have also included a discussion of alternative methods (to Knabe’s one) to proof the existence of an energy gap, and clarified Remark 3 in App. B accordingly.
We would like to reiterate that our main aim is to provide a unified framework to discuss several frustration-free models recently discussed in condensed matter physics. We achieve this by applying Witten’s conjugation arguments, which we in this way also elucidate to the condensed matter community. We believe that while this method is not new (eg, known in the MPS community), it is still fairly unknown in condensed matter physics. (This is also why we decided to keep the presentation in a notation closer to the condensed matter community.) In particular, we use this framework to clarify the interrelations between different models, as is now also highlighted by the revised title of our manuscript.
Regarding the correlation function (70) we would like to note that this has been included merely to show the general strategy to obtain such results, not because the distant-independent result (70) in itself is of physical interest. There are of course various local and non-local correlation functions one can consider, and which ones are relevant ultimately depends on the underlying setup at hand, eg, two-point spin correlation functions, Green functions of (para)fermions, etc. These correlation functions can be obtained by the reader by adapting the steps leading to (70).
List of changes
-revised the title
-changed “extend Witten’s conjugation” to “apply Witten’s conjugation” in the abstract
-revised 3rd paragraph of the introduction and added Refs. 35-37, 41
-slightly revised the wording in the 4th paragraph of the introduction
-added a new paragraph (new 6th) in the introduction to discuss Witten conjugation in the context of MPS states and alternative (more sophisticated) methods to prove the existence of an energy gap
-revised the wording in the paragraph after the proof of Theorem 1
-extended the discussion on the energy gap in the last paragraph of Sec. 5.2
-extended Remark 3 in App. B.1
-corrected some typos
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2021-7-7 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202104_00026v1, delivered 2021-07-07, doi: 10.21468/SciPost.Report.3209
Report
Since this is presumably the last round of reviews, I will try to
focus on the big picture.
Overall, I think the value of the manuscript stems primarily from the
comprehensive treatment of a large class of seemingly different models
on a unified footing, as some kind of "encyclopedic" paper (Even
though it is not encyclopedic in the sense that there is likely more
models of this kind, but this is not a point of criticism.), while
each of the individual treatments itself is quite elementary and
"low-hanging fruit", and could well be posed as an exercise problem in
a master-level course.
I think this it is absolutely fine and doesn't speak against the paper
in any way. There is nothing wrong with harvesting low-hanging fruit
in a comprehensive fashion. However, I feel it is *not* done in a
comprehensive fashion. Half of the fruit is still on the tree, and it
is not that it can only be reached by climbing high up in the tree.
Still, this would be ok if it were crystal clear from reading the
manuscript that there is still half the fruit on the tree, and it can
be reached by simply stretching one's arm. However, my feeling is that
the manuscript tells other people looking for fruit "Yes, there is
still some apples on the tree, but they are super high up, so you must
really have no fear of height if you want to reach them."
As I said previously, this perpetuates the legend that these type of
1D systems are hard to deal with, and I think this is not ok: People
in the field should be made aware of the best tools - in particular in
an encyclopedic article - since then they can apply them, and the ease
or hardness of applying the tools should not be obfuscated.
To list two very clear examples where I think the manuscript falls
short in that regard (unless I missed a very clear edit in that
direction):
1) When talking about the correlation function, there is no mention
whatsoever that using MPS & transfer matrices, one can obtain the
exact form of the correlation function, and there is an exponential
decay. This is a straightforward procedure, in particular if done
numerically. E.g., adding a plot of the correlation length would have
been minimal effort.
2) When talking about gaps, the authors state regimes where they can
bound the gap (and they give bounds). However, there is no mention of
the fact that the parent Hamiltonian of *any* injective MPS has a
unique ground state + gap. Injectivity is a property which is easy to
check (even analytically, e.g. by computing a determinant), generic,
and will - for a smooth interpolation, as the ones considered - only
break down at singular points (unless the whole path is non-injective,
i.e. it has a degenerate ground state). Even more, even for
non-injective cases, there is always a gap above the ground space.
(Intuitively, this is linked to the fact that correlations always
decay exponentially.) This is one of the key results of Refs. [32]
(FNW) and [49] (Nachtergaele). (There is Remark 3 in Appendix B.1, but
it gives precisely the idea that this is "more abstract", while in
fact checking injectivity is less work than computing the Knabe
bound.)
I think it is crucial that these things are stated clearly, as well as
that they are clear from elementary properties of the state, rather
than hiding them behind stating that (paraphrased) "similar things can
be done using MPS", and later making it sound like all this gives is
that it is much more technical and gives worse bounds on the gap (as
in Sec. 6.5).
Overall, I think it is an editorial decision whether the the
"encyclopedic" character of the overview of models compensates
sufficiently for the fact that there is still many low-hanging apples
left on the tree.
However, I feel somewhat strongly that this fact (that there are more
apples, and they are easy to reach) should be made more explicit. I
certainly dislike the idea of being told in the future by people that
they did not apply elementary MPS techniques to problem X because
applying MPS to X is hard, as can be seen by the fact that the present
paper did not do it, even though they were aware of the applicability
of MPS techniques.