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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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Authors (as registered SciPost users): Kohei Fukai · Balázs Pozsgay
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-09-25 10:02
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.

Current status:
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Reports on this Submission

Anonymous Report 3 on 2023-11-17 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2309.07472v2, delivered 2023-11-17, doi: 10.21468/SciPost.Report.8141


1- Clear conjecture supported by careful numerics.
2- Paper clearly written.


1- The results are not totally unexpected given the existing literature.


In this paper, the authors study correlation functions in models build from representations of the Temperley-Lieb algebra, in particular the Potts model, the golden-RSOS chain, or the trace representation. As is well-known, the spectrum of such Hamiltonians is contained in the XXZ spectrum. However degeneracies differ, which might affect correlation functions.

On the XXZ side several important results have been accumulated over the years. In particular, multiple integral representations for the mean values of local operators were established, and nice factorization properties were observed for those. It is natural to ask whether correlations functions in Temperley-Lieb generated models relate to known results for the XXZ spin chain.

To this effect, the authors present two conjectures, one regarding the form of the mean value of generalized current operators, the second one stating the factorization property of translationally invariant mean values of products of Temperley-Lieb generators. Both conjectures are thoroughly checked for several representations and eigenstates.

The paper is interesting and clearly written, I think it deserves publication. My main criticism is that the authors should motivate more some of the choices they make in their study. For instance, the focus on singlet states, discussing why shift invariance is important for the Potts model conjecture to hold, etc. 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.

Requested changes

Besides the above, I have a few of minor comments

a) Page 4, replace 'can not' by 'cannot'.
b) Page 15, replace 'twits' by 'twist'.
c) Page 12, after equation (46): 'and $\delta_a(x)$ is proved to be zero' reads awkwardly.
d) 'the' is missing in section 3.4.

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

Anonymous Report 2 on 2023-11-3 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2309.07472v2, delivered 2023-11-03, doi: 10.21468/SciPost.Report.8049


see report


see report


The paper "On correlation functions in models related to the Temperley-Lieb
algebra" by K.Fukai, R.Kleinemühl, B.Pozsgay and E.Vernier raises a two conjectures on the expression for two kinds of diagonal correlation functions, associated with singlet states of commuting charges in integrable models obtained from representations of the Temperley-Lieb algebra. The first conjecture provides a closed and explicit expression for the diagonal matrix elements of so-called "generalized current operators" in terms of an auxiliary function $\psi$ and its derivatives. The second conjecture expresses so called transitionally invariant mean values in terms of linear combinations of the same $\psi$ function and its derivatives.

The paper is in overall written in a clear fashion. There is only one issue that I'd like the authors to clarify. The claim in the conjectures states that the $\psi $ function is representation independent in the sense that for a given Eigenstate E, one should deduce this function from the one of the XXZ chain associated to an Eigenstate having the same value of local charges in the XXZ representation with the same TL parameter d.

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?

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.

Finally, I list a few minor remarks/suggestions for corrections

(16) what is n?

Ref[2] 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.

(56) should end with a dot

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).

Requested changes

see report

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

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

  • Cite as: Anonymous, Report on arXiv:2309.07472v2, delivered 2023-11-01, doi: 10.21468/SciPost.Report.8033


The authors study mean values of local operators generated by the Temperley-Lieb (TL) algebra underlying several integrable lattice models. The Hamiltonian $H=Q_2$ of these models is given as the sum of TL generators $e_j$ in different representations. This allows to obtain the commuting charges $Q_{\alpha>2}$ within the boost operator formalism. From these charges generalized current operators $J_{\alpha\beta}$ can be constructed which describe the flow of the charge densities $q_\alpha$ time-evolved by $Q_\beta$.

The best studied among these models is probably the XXZ spin chain. For this model short range correlation functions have be computed using their factorization property: this allows to express them in terms of products of the Taylor coefficients of two (two-point) functions $\omega$ and $\omega'$, called the 'physical part' depending on the model and eigenstate, and an 'algebraic part', depending only on the TL parameter $d$.

Focusing on singlet states of the local charges in the TL algebra or, equivalently, operators invariant under the action of the quantum group $U_q(sl(2))$ in the case of the XXZ spin chain the authors formulate two conjectures for the physical and algebraic part of the mean values, respectively. These conjectures connect the correlation functions in general TL models with the known results for the XXZ spin chain:
- the physical part of the mean values of generalized current operators is given by the Taylor coefficients of a function $\psi(x,y)$ related to the two-point function $\omega$. As shown in the earlier works [35-37] $\psi$ can be expressed in terms of the Bethe roots parameterising the eigenstate of the XXZ chain subject appropriately chosen twisted boundary conditions with the same values of local charges and with the same TL parameter $d$.
- the algebraic part for the states covered by the conjectures depends only on the TL parameter $d$ (but not on the eigenstate or the TL representation underlying the specific model).

The authors have checked their conjecture numerically for the translationally invariant expectation values of operators generated from $e_1,\dots,e_4$.

The conjectures connect mean values in singlet states of certain local operators generated by TL generators for models based on different representations of the TL algebra. This allows to use the known results from the theory of factorized correlation functions for the XXZ spin chain to the related correlators in the Potts model, an RSOS model (giving the golden chain in the Hamiltonian limit) and the trace representation of the TL algebra.
This observation extends the use of the factorization for a class of correlation functions beyond the six-vertex model. Moreover, the particularly simple simple form of the mean values of generalized currents may be useful in extending this this approach to integrable models beyond the ones considered in this paper.

I recommend to accept this paper for publication in Scipost Physics, after the points listed below have been addressed:

1- I suppose that Eq. (67) holds for singlet eigenstates $|E\rangle$ only (and the second function $\omega'$ enters in other states). If this is so the authors should emphasize this point in Conjecture 2. If not this should be explained.

2- The mean value $\langle e_1 e_4\rangle$, Eq. (72), appears to diverge for the Ising case ($d=\sqrt{2}$). The authors should discuss whether there is there any physics in this apparent divergence. Maybe this divergence is cancelled by an identity satisfied by the physical part of the expression (as observed in Refs. [9,53]).

3- Can the authors say which parameterisation (i.e. twist) of the XXZ model with $d=2\cos\pi/5$ gives the eigenvalues #2 and 3 for the golden chain (Table 9)? Similarly for the XXZ model with $d=2\cos\pi/6$ and the eigenvalues #1 and 3 for the $Q=3$ Potts model (Table 11)?

4- Caption of Table 10: should read 'current mean values of the GOLDEN CHAIN representation' instead of Potts.

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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