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

Submission summary

 As Contributors: Jean Michel Maillet · Giuliano Niccoli Arxiv Link: https://arxiv.org/abs/1907.08124v3 (pdf) Date accepted: 2020-08-26 Date submitted: 2020-08-12 12:33 Submitted by: Maillet, Jean Michel Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Quantum Physics Approach: Theoretical

Abstract

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.

Published as SciPost Phys. 9, 060 (2020)

Dear Editor,

We found very encouraging the overall positive description of our manuscript and results presented in the three referee’s reports. We would like to thank the referees for their attentive reading and understanding of the manuscript. Their numerous remarks, suggestions and identification of typos have given us the opportunity to improve the presentation of our paper and have been mainly all implemented in the current version of the manuscript. This is in particular the case for suggestions on the AdS/CFT framework and associated references that we have given in our introduction for completeness, but that we explicitly admit does not belong to our main field of expertise. The answer to referee’s remarks and suggestions is given together with the list of changes done in the improved version of our paper.

Sincerely yours,

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

List of changes

# Answer to Referee 2 remarks and suggestions:

1. We have added the two references cited in the point 1 of the report, plus a reference to a further paper of Mukin et al. Moreover, we have modified the last paragraph at page 6 and added the footnote 6 to make more explicit the difference between SoV and the Bethe Ansatz methods. The main point is that SoV is not an ansatz but a constructive resolution method; the form of the wave functions of the transfer matrix eigenvectors in the SoV basis is not an ansatz, rather, it is completely characterized by the SoV basis itself. So the main problem to solve in our approach is the construction of such SoV bases. This makes the proof of the completeness of the spectrum description very clear and even easy in general as this is mainly a built-in feature of the SoV approach. One has just to properly identify closure relations (e.g. the quantum determinant condition for the fundamental non-supersymmetric representations) that enable to act with the transfer matrix on the constructed SoV basis in a close form. Then all the transfer matrix eigenvalues are solutions of these relations and vice versa one can show that all the nonzero solutions to this system define transfer matrix eigenvalues and nonzero eigenvectors by computing the action of the transfer matrix on the constructed factorized vectors in the SoV basis. Note that for the supersymmetric case we haven’t yet computed explicitly these actions for general twists, and this is why it imposes us to keep our claim on the spectrum as a conjecture.
Instead if one uses ansatz methods, one has to first show that the constructed potential eigenvectors are indeed nonzero. In NABA this is a priori a non-trivial task. For example, to do so and to our understanding, one establishes the isomorphism between the “good” Bethe ansatz solutions and the solution of the QQ-functional equations. Then one has to show that the ansatz is complete; i.e. that all the transfer matrix eigenvectors have the form given by the Ansatz. This is generally done by counting the number of solutions, which must coincide with the dimension of the quantum space if the transfer matrix is diagonalizable. This counting is not required in SoV, because we know that the factorized form of the transfer matrix eigen-wavefunction applies to any transfer matrix eigenvector by the construction of the SoV basis (where the main point is to prove that it is indeed a basis of the Hilbert space).

2. We have taken into account the point 2 of the referee by adding the footnote 8.

3. We have implemented all the suggestions and remarks of the point 3 of the referee.

4. We have added the reference math-ph/1401.7691 to our comment on the su(2|2) symmetry at the beginning of page 5.

5. We have added the citation to paper hep-th/1501.04651.

6. We have explicitly stated “large inhomogeneities limit”.

7. To our knowledge, the quantum spectral curve is a notion and terminology introduced by Sklyanin, see for example hep-th/9212076. As explicitly written in the abstract of Sklyanin’s paper, “the canonical coordinates and the conjugated operators are constructed which satisfy the quantum characteristic equation (quantum counterpart of the spectral algebraic curve for the L operator)”. Hence, the characterization of the spectrum in SoV is given by the quantum counterpart of the spectral curve of the monodromy operator. Note that the canonical coordinate operators are the operator zeros of the Sklyanin’s B-operator, while the (exponential of) canonical conjugated operators are the shift operators on the spectrum of the canonical coordinates. In general, we write the quantum spectral curve in its coordinate form, i.e. our quantum spectral curve can be seen as the matrix element of the Sklyanin’s one between a transfer matrix eigenstate and an SoV basis element, when Sklyanin’s SoV applies, otherwise our results generalize Sklyanin’s ones. However, in general, the fact that we can prove that the transfer matrix has simple spectrum and is diagonalizable allows us to rewrite these quantum spectral curves at the operator level, involving the quantum spectral invariant operators (i.e. the transfer matrices) and just one Q-operator.

8. In fact, for the special non-invertible twist for which the quantum spectral curve (3.26) is written, we have that the $\phi_t(\lambda)$ are directly related to the eigenvalues of the $Q_2(\lambda)$ operator, as evidenced by the equations (3.54) and (3.58). However, there is in general a non-trivial normalization to consider in (3.54). Moreover, one should remark that while the quantum spectral curve (3.26) holds only for the special non-invertible twist the formula (3.29), it can be a priori extended to a general twist with nonzero $k_1$ and with $\bar\alpha=k_1$. This can be argued rewriting the transfer matrix eigen-wavefunctions in terms of the eigenvalues of the $Q_1(\lambda)$ operator by the NABA expression (3.45). For these reasons we have decided to change the notation of the zeros in $\nu$ in (3.25).

9. We agree with the referee and we have added the footnote 20.

# Answer to Referee 3 remarks and suggestions:

Concerning the link to QQ-systems, please see above comments about the relation between SoV and other methods in our answer to referee 2 (points 1 and 8). In general, QQ-systems provides necessary conditions for the eigenvalues (in the case they are derived algebraically from the R-matrix structure of the model), but it is a non-trivial task to determine which sets of solutions to such QQ-systems indeed correspond to true eigenstates. The SoV method is designed to solve such a problem. We have described the link in particular cases (see point 8 in the answer to referee 2) in the section 3.2.2. See also Appendix A.

We have also implemented the citations and clarifications that the referee has described in the section “Citation” of his report.

1. We agree with the referee and changed for the expression « twisted boundary condition » in the new version.

2. Indeed, we did not make a good distinction between the abstract algebra and its fundamental representation over C^(M|N) when introducing it. We have separated the two equations and we hope it's now clearer.

3. We have transposed most of the usual notations from the non-graded case to the graded one. For example, we are using « [ , ] » for the graded commutator, rather than the also common « [ , } ». For this reason, we keep \dagger for the Hermitian conjugation, which requires an adapted definition when being extended to a tensor space. We have added equations (2.18), (2.20) and (2.21) to clarify and illustrate the original definition (2.19).

4. The relation RKK = KKR is true for more general K matrices, but this requires introducing the supergroup GL(M|N) in details. However, since we had no use of non-diagonal twists in the present paper, we decided not to make the introductory content heavier by describing it there.

5. We agree with the referee and have modified the text accordingly.

6. We agree with the referee and have modified the text accordingly by clarifying the correspondence.

7. We agree with the referee and have modified the text accordingly.

8. The relation RKK = KKR encodes the GL(M|N) invariance of the R matrix and as such is non-trivial. It can also be interpreted as the Yang-Baxter relation for K with a trivial, one-dimensional, quantum space. However, it is a necessary requirement for the twist K such that the monodromy matrix with twisted boundary conditions given by K still satisfies the Yang-Baxter algebra.

9. We thank the referee for pointing out the poor introduction of the notation \bra{S,a}. In fact, we have reworded the statement of the theorem 3.1, as well as part of its proof relying now on the equation (2.91). We hope it’s now clearer how this object appears and how it is involved in the calculations.

10. We have thought that, being subsections of the section 3.2, the fact that the statement refers to non-invertible special case was evident. However, we agree with the referee that it is better to clarify this point. We have modified accordingly the titles of sections 3.2.1 and 3.2.2.

11. The transfer matrix is invertible in the inhomogeneities as by the reconstruction of the local operators [87,88], it is known that in the inhomogeneity $\xi_n$ this transfer matrix just coincides with the local matrix K at the site n dressed by the product of shift operators along the chain. So, it is invertible as all these operators are invertible. We have added a footnote to explain it.

12. We expect that the claim in the Conjecture 3.1 holds always for a nonzero K matrix. In the future we should be able to prove it for any twist for which it is possible to construct an SoV basis, i.e. for K being a simple matrix. Such proof by continuity arguments should lead to the proof that Conjecture 3.1 holds for almost any choice of the matrix K, i.e. up to a sub-variety in the space of the parameters of the these gl(M|N) matrices. So, there still remain some steps to prove it. We have added a footnote on this.

13. Indeed, there are some technical details that have to be developed to prove Conjecture 3.1 and, as mentioned above, we intend to do it in a further paper. The main missing point is the detailed computation of the transfer matrix action on a state who’s factorized wavefunctions in the SoV basis are written in terms of solution to the system of equations (2.108) and (2.109). The difficulty here is that the closure relations are of a higher degree compared to the standard quantum determinant relations we used in the gl(n) case. This closure relation is required to prove that these states are always transfer matrix eigenstates. Currently we have given only an argument on how this should be the case.

14. We agree with the referee and we have modified the text accordingly.

15. We agree with the referee and we have modified the text accordingly.