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Correlation functions by Separation of Variables: the XXX spin chain
by G. Niccoli, H. Pei, V. Terras
- Published as SciPost Phys. 10, 006 (2021)
|As Contributors:||Giuliano Niccoli · Véronique Terras|
|Arxiv Link:||https://arxiv.org/abs/2005.01334v3 (pdf)|
|Date submitted:||2020-11-24 06:42|
|Submitted by:||Terras, Véronique|
|Submitted to:||SciPost Physics|
We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.
Published as SciPost Phys. 10, 006 (2021)
Author comments upon resubmission
We would like to thank the referees for their attentive readings, clarification requests and for pointing out some further existing literature. We have implemented some modifications on the text of our manuscript to take them into account, mainly in the introduction and the conclusion. Let us answer separately to the two referees as it follows.
Answer to Referee 1:
1. We agree with the referee about the relevance of the extension of our SoV method to compute correlation functions to more advanced models as those associated to the higher rank cases. We have added in our Conclusion some direct comments on this and on the existing literature which defines the basis for such an extension.
The referee points out one of the central issues that has to be solved in the higher rank case in order to extend our SoV approach to correlation functions, i.e. how to compute the action of local operators on transfer matrix eigenstates in SoV framework. Thanks to the reconstruction formula  this is mainly equivalent to the computation of the action of the monodromy matrix elements on the SoV basis. In , the fusion equations are indeed the main ingredient to compute the action of the transfer matrix on a separate state and to fix its form in order to be a transfer matrix eigenvector. It is then natural to expect that the fusion of the monodromy matrix themselves should play a role in this “out of diagonal” type of action computations. Some ideas are currently under investigation in this direction but they are still too premature to comment openly on them.
Another important point is the derivation of efficient scalar product formulae of determinant type. The results in  are going in this direction, see also [128-129]. In particular, we would like to mention that in , a special choice of the convector/vector SoV bases has been defined which manifests a strong analogy in the SoV measure to the one for the rank one case and leads to Slavnov’s type determinant formulae for scalar product of separate states with transfer matrix eigenstates. Such results when accompanied by the computation of the action of local operators are at the basis of the generalization of our SoV method toward correlation functions.
Said that, we want to stress that the main aim of the present manuscript is to introduce and illustrate, for a simple set of models, a very much needed method to compute correlation functions in the SoV framework. To our knowledge suck a method is not outdated but rather a complete novelty in the literature and hopefully of potential large use. Moreover, while computing correlation functions for higher rank model would surely be an outstanding result, this is not the next natural subject of our research. Indeed, we want to stress that there is a large variety of rank one models whose correlation function are not available in the literature and whose computations would represent a fundamental achievement. This is for example the case of the open XXX/XXZ/XYZ spin chain with general boundary conditions.
2. We agree with the referee suggestion and we have added some text to explain it. In fact, we call them in that way because they have a Bethe Ansatz form but they are not derived by Bethe Ansatz. Indeed, no ansatz is done on them and they are just a simple rewriting of the SoV separate state description of transfer matrix eigenvectors. 3. We have implemented the text suggestion of the referee.
Answer to Referee 2: We agree with the referee that our introduction missed the discussion of some relevant literature and we have modified accordingly our manuscript. 1. We have listed the essential ingredients for computation of correlation functions as required by the referee. This has given us the opportunity to further stress the role of the knowledge of the density of the Bethe roots of the ground state to compute correlation functions, as first remarked in . Concerning scalar products for models without U(1) symmetries, we agree in the interest of the developments in the framework of the so-called modified ABA (MABA), and we have reported them accordingly to the referee suggestions. Concerning the interesting paper 2005.11224, let us comment that it is subsequent to our manuscript, nevertheless we have decided to take it into account. 2. At page 4-5, we have recalled the existing approaching by inhomogeneous TQ Baxter equations and the non-polynomial Q-operator with the associated literature. 3. Let us comment that we didn’t write explicitly the form of the SoV basis as it does not play any role in the computation of the correlation functions. As it is known now by the paper , we can have different representations for the same states and currently in the text we are explicitly referring to this paper for these different explicit forms. 4. The question of the referee is not completely clear to us. We can compute the action of the transfer matrix on a generic separate state. This can be done by using the SoV representation of the Yang-Baxter algebra in the SoV basis as given in page 8. It is exactly this type of computation which fix uniquely and without any ansatz the form of the transfer matrix eigenstate as given in pages 9-10, which are special instances of separate states. We do not recall these steps here as they are well known results in the literature, see for example  for a recall in the XXX case. 5. Thanks for the remarked typo, we correct it. 6. We are not at the knowledge of any further reference. 7. We have added the reference to the paper where this homogeneous TQ equation first appeared and clarified the role played in  in proving it.
Finally, from the referee reports, we would like to make some remarks.
a) While not being the main aim of our manuscripts, the results here derived are to our knowledge entirely new. They were only expected on physical basis but not proven elsewhere in the literature by any other method.
b) The scalar products of Slavnov form first appeared and have been proven for twisted XXX chains in the SoV framework in our previous paper . The scalar products in the MABA framework have a similar form to those derived in the SoV framework (i.e. there they are rewritten as the determinant of the Jacobian of the transfer matrix eigenvalues or some local deformations of it). Nevertheless, it seems that in MABA framework the description of the transfer matrix eigenvalues are always given in terms of inhomogeneous TQ-equations. This is not always the case in the SoV framework. On the one hand, this makes complicate a direct connection between these formulae. On the other hand, to our knowledge, the analytic identification of the ground state at the thermodynamic limit is a very much complicate and still unsolved task for the inhomogeneous TQ equation (at least in terms of a density function solution of an integral equation in the thermodynamic limit with some control on the finite-size corrections, as from usual Bethe equations). This is one of the fundamental ingredients to analytically compute correlation functions, which is still missing in the literature for this inhomogeneous case.
Nevertheless, when considering the XXX spin chains with general quasi-periodic boundary conditions in the SoV framework, we don’t need to use such an inhomogeneous TQ equation (we could of course, but it would be counter productive for our purpose), since it is easy to rewrite the SoV spectrum and eigenstates in terms of solutions of a usual (homogeneous) TQ equation, as done in . From which we can easily obtain a simple description of the ground state in the thermodynamic limit in terms of a density function and so implement as described in the manuscript the computation of correlation functions.
List of changes
See the author comments which list the changes made according to each referee's suggestions
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2020-12-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2005.01334v3, delivered 2020-12-09, doi: 10.21468/SciPost.Report.2274
I am satisfied with the changes the authors have implemented, in particular for their nice discussion regarding the possible generalisation of their techniques to higher rank models and for clearing up some potentially confusing terminology used in the paper.
I am now happy to recommend the paper for publication.