Thermodynamic limit of the two-spinon form factors for the zero field XXX chain

Submission summary

 As Contributors: Nikolai Kitanine · Giridhar V. Kulkarni Arxiv Link: https://arxiv.org/abs/1903.09058v2 Date accepted: 2019-06-11 Date submitted: 2019-05-21 Submitted by: Kitanine, Nikolai Submitted to: SciPost Physics Discipline: Physics Subject area: Mathematical Physics Approach: Theoretical

Abstract

In this paper we propose a method based on the algebraic Bethe ansatz leading to explicit results for the form factors of quantum spin chains in the thermodynamic limit. Starting from the determinant representations we retrieve in particular the formula for the two-spinon form factors for the isotropic XXX Heisenberg chain obtained initially in the framework of the $q$-vertex operator approach.

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Published as SciPost Phys. 6, 076 (2019)

We would like to thank the referees for several useful comments and also for the detected typos, we followed most of the referees suggestions in the new version. We would like also to reply to some of questions and suggestions.

Referee 1:

1. The method described here can be applied for the XXX and XXZ case. The additional $\mathfrak{su}(2)$ symmetry is a particularity of the XXX case and it brings some technical difficulties absent in the XXZ case. To deal with these difficulties we apply the Foda-Wheeler modification of the Slavnov formula, it permits us to compute some integrals and helps a lot for the computation. We agree that this fact can be mentioned in the introduction, however we think that this purely technical part it is not that essential for this approach to be mentioned in the abstract.

2. We thank the referee for reminding us some extra papers on thermal form factors, in particular Dugave et al, J. Stat. Mech. (2015) P05037 is highly pertinent to the present paper (due to the discussion on the comparison of $q$-vertex operator and ABA results). We are also aware of the large literature on the form factors for integrable quantum field theories, but we think that this paper is dealing only with the spin chains and we have mostly restricted our reference list to these systems.

3. We thank the referee for raising this important question on the scaling behaviour of the form factors for zero and non-zero magnetic field. The non-integer scaling was obtained for the finite magnetic field case and it plays an important role when the excitations (particles and holes) are close to the Fermi boundaries. In this paper we consider the zero magnetic field case (Fermi boundaries are at infinity in the thermodynamic limit) with holes in the bulk (far from the edges of the Bethe roots distribution for finite $M$). In this case it is easy to see that the boundary values of the shift function which give the non-integer scaling (see for example eq. (3.14) in N. Kitanine et al J. Stat. Mech. (2011) P05028) are vanishing and lead to integer powers of $M$. We agree that this discussion can be included in the paper.

4. Corrections of order $1/M$ are essential for the computation of determinants of matrices of size proportional to $M$. For this reason these terms are carefully taken into account (holes contributions). We claim after eq. (3.11) that all the corrections sub-leading to $1/M$ can be neglected, it follows from the development of $\det(I+xA)$ as a power series in $x$.

6. We are not sure that a picture (in the simplest two-spinon situation) would help much in this context.

7. The denominator in eq. (2.33) contains two norms of two states involved in the form factor, there is no particular factorisation.

Referee 2:

We agree that the bulk assumption is the most subtle point of the presented method and for this reason we wanted first test it by obtaining a well established result. We also agree with Frank Göhmann that the proof of this assumption (or a weaker one explaining why some finite size corrections are not pertinent in the thermodynamic limit) could be tried through the analysis of non-linear integral equations. However this analysis goes far beyond the scope of the present paper.

Referee 3:

We would like to thank Frank Göhmann for several important remarks, in particular for reminding us to cite the paper [45] (in the new version) where effectively a very similar approach is used to compute the ratio of two matrices.

We have deliberately chosen a rather short introduction for this (rather technical) paper covering the simplest form factor reserving a more detailed analysis of the context for forthcoming publications treating states involving complex roots. However we have added some important references which were evidently missing.

List of changes

1. Introduction: slightly extended, several references added as suggested by referees
2. Section 3: citation of the paper [45] is added (as suggested by referee 3), some explanation on the scaling behaviour is provided at the end of subsection 3.2 (as suggested by referee 1), some explanation of the computations are given after eq. (3.52)

Several typos corrected throughout the paper

Submission & Refereeing History

Resubmission 1903.09058v2 on 21 May 2019
Submission 1903.09058v1 on 28 March 2019

Reports on this Submission

Report

I remain of the idea that the paper is still too technical and that a more general introduction on spinons and form factors expansion of correlation functions was needed. But overall I think that now the paper is ready for publication.

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Report

In my previous report, I have already recommended the paper for publication. With the improvements of this version, I recommend the paper for publication as it is.

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Report

I think there is not much to add to my previous report. The authors have further improved the manuscript by taking account of the referee's suggestions. The manuscript should now be published as it stands.

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