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On correlation functions for the open XXZ chain with non-longitudinal boundary fields : the case with a constraint

by G. Niccoli, V. Terras

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Submission summary

Authors (as registered SciPost users): Giuliano Niccoli · Véronique Terras
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-02-16
Date submitted: 2024-01-26 22:06
Submitted by: Terras, Véronique
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


This paper is a continuation of [1], in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in [1].

Author comments upon resubmission

Dear Editor,

We would like to sincerely thank the three referees for their attentive study of our manuscript, for their requirements of clarifications, suggestions and list of grammar misprints.

In the new version of our manuscript, we have corrected the list of typos noticed by the referees, and implemented their grammar suggestions. We have also made some other minor corrections, see our detailed answer below.

1) On the referee 1 report.

  • About the request 1: We have modified the first paragraph of Section 4 to take into account the first referee request about the description of our basis of local operators. We have modified, in particular, the equation (4.1), where now we have defined the 4 local gauge transformed operators at a given site n (instead of their product). For clarity, we have also added, in footnote 7, their explicit expressions as linear combinations of the usual elementary operators at site n. Then, the elements of our basis (4.2) are simple tensor products of these local operators (4.1). The only peculiarity/complexity of our basis lays in the fact that the parameters b and bar b are functions by (4.3) of the full m-tuples of epsilons and prime epsilons. Following the referee request, we have also added a footnote where this dependence is written explicitly for m=1,2. One has to mention that our basis led to simple actions on the separate states which is indeed very similar to the ungauged action in our paper [20] (diagonal case), we have also added a footnote with this comment (footnote 6). This is a strong simplification, which enabled us to compute the action explicitly despite the combinatorial complexity induced by the use of the vertex-IRF transformation. This is of course the reason of the choice of this particular basis.

  • About the request 2: It is clear that we are not doing a full analysis of the ground state of the boundary spin chain here. We have made this even more transparent by modifying one sentence in the paragraph after formula (6.7) and adding an additional sentence in the Conclusion. Such an analysis is quite involved, since we have to distinguish many cases according to the different values of the boundary parameters, and we have even to compare what is happening in two different sectors if we trust the completeness conjecture of [14]: see for instance the analysis done in [41] for the diagonal case, knowing that here it is much more involved since there are much more cases to distinguish. Such an analysis of the ground state, if done properly and precisely, would probably deserve a separate publication. Therefore, we are doing some assumptions here: as stated in page 19, we assume that we are in a configuration of Bethe roots such that the ground state is indeed described by an infinite number of real Bethe roots, leading to a density function (6.4) solution of (6.5) and (6.6), with possibly some additional boundary roots of the form (6.7). Under this assumption we obtain our result (6.8). We nevertheless point out just before these assumptions that the thermodynamic analysis of the ground state can in principle be implemented as in [8,9,41], if the latter is in the sector N/2, or even in a sector which differs from N/2 by a finite number: this will lead to the ground state density function (6.4). Of course, depending on the precise sector and on the value of the boundary parameters, the fine structure of the ground state may be very different (with possibly, among other possible effects, the presence of some holes, as mentioned by Referee 2). The presence of a finite number of holes should however not modify the leading order of the result in the thermodynamic limit, and we have added a footnote (footnote 12) mentioning it. In principle, the presence of a finite number of additional roots should not modify the leading order either, except if these roots correspond (in the thermodynamic limit) to some singularities of our expression: this is precisely what happens for the boundary roots of the type (6.7), and this is why they are especially important for the statement of our result, as in the diagonal case [20,41]. The first paragraph of section 6, instead, is meant to clarify that, under the completeness conjecture of [14], we have that if M is of order N/2 then the same is true for M’=N-1-M so that the ground state has a number of roots of order N/2, independently if it is in the sector M or M’, and so the ground state analysis which is sketched just after that applies.

We have slightly modified a few sentences (and added footnote 12) in these paragraphs so as to make all this clearer.

2) On the referee 2 report.

We understand the referee statement about technical complexities of our manuscript, although this is somehow inevitable due to the technical complexity of the research subject. We will try to make our best, in our future publications, to simplify somewhere the exposition by giving concrete examples.

About the referee comment on Theorem 4.1, we would like to mention that the form of this result mimics similar results that have been previously obtained for the correlations functions of the spin chain with different types of boundary conditions. As mentioned in Remark 2, it has of course the same formal form as in our previous work [1] with the special boundary condition (4.17). More remarkably, it has mainly the same structure as in Proposition 5.2 of [20], which stated the action of the natural basis of local operators on usual (ungauged) Bethe vectors for diagonal boundary conditions. Here, the main difference is in the last two lines of (4.12), when compared to (5.15) of [20], where we can read of explicitly the dependence from the b- and bar b-parameters of our current local basis. It is this relative simplicity that allows us to compute correlation functions in our current general boundary conditions.

We also agree that it is not clear how much practical may be our formulae (and our final result) for implementing a numerical set-up. This is only the first step towards a full understanding of the correlation functions of the model. Nevertheless, considering the complexity of the problem of dealing with this kind of boundary conditions, we think it is important to understand first how to generalize the results which are known in the diagonal case [20] before trying to go further, and this is the purpose of the present work. A possible further and more concrete study would be for instance to try to generalize the results of [41] to this case. For more general numerical applications, the difficulty in the open chain (even for diagonal boundary conditions) is that, contrary to the periodic case [93], we lack a compact and manageable representation for the form factors as soon as we go away from the boundary. Trying to overcome this difficulty is an interesting and very challenging problem, but it should obviously be addressed first in the simpler diagonal case.

3) On the referee 3 report.

The referee 3 points out as a weakness the fact that we restrain ourselves to the standard TQ-equation. In fact, as the referee mentions, our main motivation to consider the case with one constrain is due to the fact that the problem can be reduced to the standard TQ-equation for which the thermodynamic analysis can be addressed on more standard basis. Instead, the unconstraint most general integrable boundary conditions naturally lead to a TQ-equation with one additional (inhomogeneous) term, once one asks for trigonometric polynomial Q-functions. There exist some analyses of this most general case, but it is clearly much more difficult to deal with than the constraint case. In particular, it is not clear how to deal with the thermodynamic limit in that case, since the Bethe roots can a priori not be described by a simple density function as in the constraint case. But we agree with the referee that considering this most general case is an interesting – although difficult – open problem, which is commented as such in the conclusion.

List of changes

- small grammatical corrections, as requested by the referees
- slight modifications at the beginning of section 4 so as to better clarify the definition of the basis of local operators (4.2), as requested by referee 1
- slight modifications of section 6, just after (6.3) and around (6.7), and one sentence added in the conclusion, see our detailed answer to the request 2 of referee 1

Published as SciPost Phys. 16, 099 (2024)

Reports on this Submission

Anonymous Report 2 on 2024-2-10 (Invited Report)


I am happy with the authors response to my report, and look forward to seeing concrete examples in their future publications. I recommend the paper for publication.

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

Anonymous Report 1 on 2024-2-9 (Invited Report)


Authors responded to my remarks, I recommend to accept the paper

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

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