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Separation of variables bases for integrable $gl_{\mathcal{M}|\mathcal{N}}$ and Hubbard models

by J. M. Maillet, G. Niccoli, L. Vignoli

Submission summary

As Contributors: Jean Michel Maillet
Arxiv Link: (pdf)
Date submitted: 2020-04-28
Submitted by: Maillet, Jean Michel
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Mathematical Physics
Approach: Theoretical


We construct quantum Separation of Variables (SoV) bases for both the fundamental inhomogeneous $gl_{\mathcal{M}|\mathcal{N}}$ supersymmetric integrable models and for the inhomogeneous Hubbard model both defined with quasi-periodic twisted boundary conditions given by twist matrices having simple spectrum. The SoV bases are obtained by using the integrable structure of these quantum models,i.e. the associated commuting transfer matrices, following the general scheme introduced in [1]; namely, they are given by set of states generated by the multiple action of the transfer matrices on a generic co-vector. The existence of such SoV bases implies that the corresponding transfer matrices have non degenerate spectrum and that they are diagonalizable with simple spectrum if the twist matrices defining the quasi-periodic boundary conditions have that property. Moreover, in these SoV bases the resolution of the transfer matrix eigenvalue problem leads to the resolution of the full spectral problem, i.e. both eigenvalues and eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a trivial overall normalization) eigenvector whose wave-function in the SoV bases is factorized into products of the corresponding transfer matrix eigenvalue computed on the spectrum of the separate variables. As an application, we characterize completely the transfer matrix spectrum in our SoV framework for the fundamental $gl_{1|2}$ supersymmetric integrable model associated to a special class of twist matrices. From these results we also prove the completeness of the Bethe Ansatz for that case. The complete solution of the spectral problem for fundamental inhomogeneous $gl_{\mathcal{M}|\mathcal{N}}$ supersymmetric integrable models and for the inhomogeneous Hubbard model under the general twisted boundary conditions will be addressed in a future publication.

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Reports on this Submission

Anonymous Report 2 on 2020-6-12 Invited Report


1- The paper is very well written, and largely self-contained
2- It contains new relevant results on a topic of renewed interest in the integrable systems community. The potential applications of these methods range from condensed matter to AdS/CFT
3- The bibliography and review of the literature is overall comprehensive and balanced.


1- I feel that some references should be added to other approaches to the completeness problem, and a short discussion on how they compare to the approach of the authors (more comments on this below).
2-Often, the authors refer to arguments in their previous works to shorten the proofs of theorems. The references are very precise, however occasionally I feel that a few more words of explanation should be offered to help the reader follow the logic (I will point out one example below).
3- The review in the introduction of developments in AdS/CFT integrability is in my view incomplete (more comments below).


The paper deals with the separation of variables method in quantum integrable systems, continuing a series of important develoments in the area in the last years, by the authors and other groups.
The paper addresses the construction of separated variables bases for
supersymmetric rational spin chains at any rank, as well as the Hubbard model. This is done applying a method recently developed for the non-supersymmetric case by two of the authors in [1] , whereas the separated variables basis is built by repeated action of the transfer matrix evaluated at special points on a reference state.
After constructing the basis, the authors use it to prove some spectral properties of the transfer matrix, such as simplicity of the spectrum and diagonalizability (under some conditions on the twists).
They also emphasize the role of a nice and simple quantization condition for the transfer matrix eigenvalues (generalizing the "quantum determinant" of the bosonic case), and conjecture that it gives a complete characterization of the spectrum. They prove this statement for a particular choice of twists and superalgebra, providing an explicit example of the construction.
The paper is very well written, and the results are relevant for developing the SoV strategy to the computation of observables in supersymmetric spin chains and AdS/CFT. I have no doubt in recommending the paper for publication. However, I kindly ask the authors to consider some minor corrections detailed below.

Requested changes

1. When discussing completeness, in particular in the Introduction and section 2.5, I would ask to include a reference to the works on completeness based on the Wronskian QQ relations. In particular, to the best of my knowledge the work math.QA/1303.1578 is considered as a proof of completeness of the Bethe Ansatz for the gl(N) XXX spin chain. I also mention the related recent work math-ph/2004.02865 which presents a proof of completeness for the supersymmetric case (of course, this work appeared after the work of the authors, therefore I leave it up to them whether to include this reference).
In general, I would really appreciate seeing a short discussion of links and differences of the authors approach with the QQ relations approach.
2. In the second paragraph on page 6 , when talking about generating states with a single, non-nested B operator evaluated at the zeros of the Q function, I would ask to add a citation to [150], where this property was first discovered and the form of B conjectured for any rank.
3. I would like some points to be clarified in the discussion of AdS/CFT integrability, and I kindly ask the authors to take into accounts the following comments. First, the AdS/CFT S matrix is not equivalent to the Hubbard R-matrix, but to two copies of the latter, multiplied by a (nontrivial) dressing phase. Second, the direct relation between the dilatation operator and the Hubbard Hamiltonian is only valid at weak coupling (and in a special sector). Third, spin chain approaches such as those in [148]-[149] are nowadays known to miss one important part of the result for the AdS/CFT spectrum, the so-called wrapping corrections, and have been surpassed by approaches based on the TBA. Finally, I think the authors should include in this discussion the fact that a full solution for the spectral problem of AdS/CFT has been proposed in hep-th/1305.1939 (this has been tested extensively and is regarded as one of the main results in this area).
4.- The study of the Hubbard model in a paper dealing mostly with supersymmetric spin chains could be motivated more strongly. I believe an explicit link exists since the R matrix of the Hubbard model has a su(2|2) invariance (this is e.g. discussed in reference [132] or in math-ph/1401.7691). It could be useful to make this comment.
5. Since the authors cite the fusion hierarchy for the Hubbard model, I point out to them the paper hep-th/1501.04651 where the hierarchy of functional relations and the related quantization conditions are discussed in the finite temperature case.
6. In the proof of Th. 2.1, when mentioning "special limits" used in the proof, it would help the reader if the authors recalled explicitly the limit of large inhomogeneities.
7. Can the authors elaborate on the name "quantum spectral curve" for eq. (3.26)?
8. I feel it would be better to name the zeros of \phi in (3.25) \mu rather than \lambda, consistently with (3.38) and (3.58).
9. This is a simple suggestion. To make contact with other SoV approaches using the B operator, it may be relevant to remark that (3.29) is a rewriting of the wave functions in terms of Q functions.

10. In (2.90) since this is a covector, should it belong to V_a^* ?
11. I believe there is a typo in (2.68), namely the signs of the shifts in the arguments of the the T functions should be reversed.
12. For notation consistency, in formula (2.36) should it be M^(K) ? Otherwise, I believe the superscript ^(K) comes in (2.43) with no previous explanation.
13. The notation \xi^(n) is used only in two equations. For clarity I would advise writing the shift explicitly in (3.59), since the definition of this notation is difficult to spot.
14. Some typos: "ortogonality" should read "orthogonality" in a couple of places. Another typo is "antisymetric" on page 7. Finally, a typo in one of the names in ref. [135].
15. Below (3.54), the degree of the polynomial \bar \phi is given as M \leq N- m_h. However for consistency with (3.25) , it looks like it should be M- m_h instead.
16. The footnotes 10 and 11 are identical.
17. A power of \lambda is missing in eq. (A.7)
18. It looks like (B.9) and (B.18) are repetitions of (B.8) and (B.17).

  • validity: top
  • significance: good
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  • formatting: perfect
  • grammar: excellent

Anonymous Report 1 on 2020-5-11 Invited Report


The separation of variables is potentially the most powerful method for investigation the off-shell behaviour of the quantum integrable models. In the paper by J.M.Maillert, G. Nicolli and L. Vignoli the problem of separation of variables is solved for models with SUSY. This case is important for studying the Hubbard model, the case considered in details in the paper, another important application can be provided by AdS/CFT correspondence. This is a highly professional and well-written paper, I do recommend it for publication.

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