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On correlation functions in models related to the Temperley-Lieb algebra

by Kohei Fukai, Raphael Kleinemühl, Balázs Pozsgay, Eric Vernier

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Submission summary

Authors (as registered SciPost users): Kohei Fukai · Balázs Pozsgay
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-12-20
Date submitted: 2023-11-30 04:54
Submitted by: Pozsgay, Balázs
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


We deal with quantum spin chains whose Hamiltonian arises from a representation of the Temperley-Lieb algebra, and we consider the mean values of those local operators which are generated by the Temperley-Lieb algebra. We present two key conjectures which relate these mean values to existing literature about factorized correlation functions in the XXZ spin chain. The first conjecture states that the finite volume mean values of the current and generalized current operators are given by the same simple formulas as in the case of the XXZ chain. The second conjecture states that the mean values of products of Temperley-Lieb generators can be factorized: they can expressed as sums of products of current mean values, such that the coefficients in the factorization depend neither on the eigenstate in question, nor on the selected representation of the algebra. The coefficients can be extracted from existing work on factorized correlation functions in the XXZ model. The conjectures should hold for all eigenstates that are non-degenerate with respect to the local charges of the models. We consider concrete representations, where we check the conjectures: the so-called golden chain, the $Q$-state Potts model, and the trace representation. We also explain how to derive the generalized current operators from concrete expressions for the local charges.

Author comments upon resubmission

We are thankful to the referees for the comments. Please find our replies below.

List of changes

To Referee 1

1. We made conjecture 2 more precise, mentioning the singlet properties. At the moment we don't know about omega' in other (non-singlet) states.
2. As pointed out by the referee, the apparent divergence of some correlators as $d\to 2$ (and, in fact, simularly for $d\to 1$) is cancelled by a cancellation of the numerator, as was previously observed in Refs [9,53]. We have commented on this fact at the end of Section 5.1.
3. We added the explanation of the corresponding twist and magnetization for the eigenstates of Golden and Potts in the
caption of Tables 9 and 11, as the referee requested.
4. We fixed the typo pointed out in the caption of Table 10.

To Referee 2

Major comments from Referee 2

Q: "First, how do the authors know that for a given TL representation with parameter d can they always find en Eigenstate on the XXZ chain side having the claimed properties?"
A: We added an explanation on the equivalence of periodic TL rep and XXZ. We also changed the Hamiltonian to the periodic one explicitly.

Q: "Second, it seems to me that the \omega function, say in (52), is the only representation dependent part in the formula (eg it changes whether one is in finite volume, infinite one, finite temperature, chosen averaging Eigenstate, etc). I think that it would be clearer if the authors managed to present their conjecture in terms of some object/quantity/formula that they directly attach to the representation for which the formula is written."
A: We added some clarification about the omega function in multiple places, explaining that it can take different values
depending on the physical situation, but the factorization is always the same. We hope it is clear now.

Minor remarks/suggestions for corrections

Q: "(16) what is n?"
A: We fixed the typo in (16), changing n -> Q.

Q: "on page 9 should be replaced by the original resut B.M.~McCoy and T.T.~Wu, "Hydrogen-Bonded Crystals and the Anisotropic Heisenberg Chain.", Il Nuovo Cimento B, 56, (1968), 311-315."
A: We added the suggested reference.

Q: "(56) should end with a dot"
A: Typo fixed.

Q: "It was unclear to me during reading whether (54) and (60) correspond to the same function ( omega was not defined explicitly in the paragraph related to the XXZ chain). I would suggest to clarify that point somewhere around (60)."
A: This is the same function indeed.

To Referee 3

Requested small changes, which are now fixed:
(a) can not -> cannot
(b) twits -> twist
(c) Kohei erased and $\delta_\alpha(x)$ is proved to be zero.
(d) we added the missing "the" in the title of the subsection 3.4.

Other comments, suggestions:

Q: "Similarly, they should at least discuss whether the approach used to show factorization in XXZ would work directly or not in the models they study. In this regard the third to last and second to last paragraphs in the conclusion feel a bit lazy."
A: We expanded the third to last paragraph in the Conclusion, mentioning a possible direction for a proof. Also, we
extended the paragraph afterwards.

Q: "For instance, the focus on singlet states, discussing why shift invariance is important for the Potts model conjecture to hold, etc."
A: We added the explanation on singlet eigenstates and non-singlet eigenstates at the end of Section 5.1, furthermore we
added some explanation about the relevance of the shift invariance to the Conclusions.

Published as SciPost Phys. 16, 003 (2024)

Reports on this Submission

Anonymous Report 3 on 2023-12-13 (Invited Report)


The authors have addressed the issues raised in my report on the original submission. The paper should be published in its present form.

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

Anonymous Report 2 on 2023-12-12 (Invited Report)


I believe the authors have addressed my concerns, I recommend publication.

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

Anonymous Report 1 on 2023-11-30 (Invited Report)


see previous report


see previous report


No more comment. I suggest to publish.

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

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