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Correlation functions by Separation of Variables: the XXX spin chain
by G. Niccoli, H. Pei, V. Terras
This is not the current version.
|As Contributors:||Giuliano Niccoli · Véronique Terras|
|Arxiv Link:||https://arxiv.org/abs/2005.01334v2 (pdf)|
|Date submitted:||2020-07-22 08:45|
|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.
Submission & Refereeing History
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Anonymous Report 2 on 2020-10-12 Invited Report
- Cite as: Anonymous, Report on arXiv:2005.01334v2, delivered 2020-10-12, doi: 10.21468/SciPost.Report.2075
1 - Reformulation of the SoV correlation functions in term of the Bethe roots
1 - Original References not always accurate
2- No discussion of the problematic of the TQ equation with an additional term to reconstruct correlation function in term of the Bethe roots.
The authors study the correlation function of the twisted XXX spin 1/2 chain by means of the separation of variables (SoV). They recover
known results about the model. Moreover they show the independence of the model to the twist in the thermodynamical limit.
The paper is of interest and deserve publication. But some points must be improved before this.
- p3 the authors could itemize the essential ingredients to study correlation function. The Slavnov formula’s can be obtain by means of the algebraic Bethe ansatz for models without U(1) symmetry (arXiv:1506.06550, arXiv:1507.03242, arXiv:1805.11323, arXiv:1906.06897, arXiv:1908.00032,arXiv:2005.11224 ). The lack of reference state is due to the lack of U(1) symmetry of the Hamiltonian, but Bethe ansatz (off diagonal or modified algebraic) still work in that case.
- p4 the authors must discuss more clearly the situation about TQ equation with an additional term (originally from Off diagonal Bethe ansatz approach) and the case with non polynomial Q function. They can gives more details about what is a Slavnov type determinant.
- p7 the authors could gives a explicit formula for the basis |h>.
- p8-9 Did the author can find the action of the transfert matrix on an arbitrary separate state ?
- p13 typo eq (4.16) K->k.
- p14 Did the author have more reference about the ground state of the diagonal twist case ?
- p16 ref  the original reference must be added
Anonymous Report 1 on 2020-10-1 Invited Report
- Cite as: Anonymous, Report on arXiv:2005.01334v2, delivered 2020-10-01, doi: 10.21468/SciPost.Report.2036
1. Explicit implementation of the SoV approach in the thermodynamic limit.
2 - Proof that correlation functions in the thermodynamic limit do not depend on the specific choice of boundary twist.
3 - Rewriting of the action of monodromy elements on separated states in terms of Bethe roots instead of inhomogeneities.
1 - Restricted to gl(2)-type models using methods which are somewhat outdated and not well suited to higher-rank generalisations.
The authors have considered the problem of computing correlation functions in the thermodynamic limit for the XXX gl(2) spin chain. They begin by reviewing the separation of variables procedure for this spin chain with anti-periodic boundary conditions and construct the appropriate separated variable states and compute the action of the monodromy matrix elements on these states. Then, they show how this action can be rewritten in terms of a family of contour integrals which can then be re-expressed as a sum over Bethe roots, which is one of the central results of the paper. Next, the authors combine their results to compute zero-temperature correlation functions. Finally the authors generalise their computations to the case of more general non-diagonal twists and nicely show that the correlation functions in the thermodynamic limit are independent of the specific choice of twist.
Overall I am impressed with the quality of the paper and it certainly merits publication, but I have a few changes which I ask the authors to consider implementing.
1 - The most important change which I request the authors implement is to add at least some discussion about possible generalisations to higher-rank. Given the recent tremendous progress in developing the separation of variables procedure for higher-rank I feel that such a discussion is crucial for the completeness of the paper. To elaborate further, here the situation is a bit simpler because the action of all A,B,C,D on the SoV states can be easily computed, since one acts diagonally, two act as simple raising / lowering operators and the remaining one can be computed from the quantum determinant. This is no longer true at higher rank where in principle the only thing one knows is the action of the transfer matricies and the associated fusion relations. Is this enough to reconsruct the action of all monodromy elements?
I should stress that I do not expect the authors to do any actual computations / derivations / proofs relating to this, but to at least comment on these potential difficulties and how they might be overcome, for example in the Conclusion section of the paper where the authors already discuss some possible generalisations to other more complicated models.
2 - Under 3.20 please define what a “generalised Bethe state” is and perhaps also explain how it is different from a usual Bethe state (which I would consider to be any eigenvector of the transfer matrix).
3 - Finally, here are some typos / grammatical errors I noticed:
“Here develop” -> “develop here”
“Like here poles” -> like poles
“In related SoV framework” -> “in a related SoV framework” or perhaps just “in the SoV framework”.
“Implement here our approach” -> “implement our approach here”
“Not necessarily solution” -> “not necessarily a solution”
“Operators zeros” -> operator zeros”
“Results than in” -> “results as in”
“One of the difficulty” -> “one of the difficulties”
“Achievement than the” -> “achievement as the”