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Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

by J. M. Maillet, G. Niccoli

Submission summary

As Contributors: Jean Michel Maillet
Arxiv Link: https://arxiv.org/abs/1810.11885v3
Date accepted: 2019-06-04
Date submitted: 2019-05-16
Submitted by: Maillet, Jean Michel
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Mathematical Physics

Abstract

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to gl(n)-invariant R-matrices in the fundamental representations. We consider lattices with N sites and quasi-periodic boundary conditions associated to an arbitrary twist K having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, i.e., from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order n. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. If the twist matrix K is diagonalizable with simple spectrum then the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter Q-operator satisfying a T-Q equation of order n, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.

Current status:
Publication decision taken: accept

Author comments upon resubmission

Dear Editor,

We are sincerely grateful to the referees for their careful reading of the manuscript, the interest shown in our work, their valuable remarks and moreover for the useful detection of several misprints.

We have implemented most of the referee’s requests. In the following, we give in detail the main modifications that we have done (besides, we also fixed some other minor misprints), and we answered the questions or remarks of the referees.

Both the referee’s 1 and 2 have asked about the homogeneous (and thermodynamic) limits. We have added a conclusion section to our paper, in particular to discuss this issue and also to give further comments on future developments. Here, let us just say that the general idea is to develop our analysis for both the spectrum and the dynamics in the inhomogeneous models and then to consider their homogeneous limit. This strategy has proven to be the most efficient in our previous works, in particular in the algebraic Bethe ansatz approach to correlation functions of quantum integrable lattice models, and we think this is also the best one in the present context. Indeed, considering first the inhomogeneous models enables one in general to give a simpler algebraic description of all the intermediate steps that would otherwise involve various (high) derivatives of the elementary objects involved. Moreover, let us remark that the characterization of the spectrum by TQ finite difference functional equation admits in general a smooth homogeneous limit. So, starting from the inhomogeneous complete characterization of the spectrum one gets the homogeneous one. However, one still has to prove that in the homogeneous limit this characterization is complete, i.e. that the full spectrum is described by the TQ equation. One way to prove it can be by direct construction of the SoV basis in the homogeneous models. This requires the proof of the non-degeneracy of the spectrum of the full set of fused transfer matrices in this limit. The other main subject which it is under analysis is the computation of scalar products and matrix elements on separate states for higher rank case. In the SoV framework for integrable quantum models related to gl(2), i.e. to rank one case, this type of analysis has been already developed with our collaborators N. Kitanine and V. Terras. Hence, the current analysis will represent a natural but non trivial extension of it.

The second referee asked also about a constructive derivation of the TQ equation in the SoV framework. We have added some comments about this in the Remark 4.1 after the Theorem 4.1 where this TQ equation is given. Let us say that from our point of view the way the TQ equation has been derived in our SoV framework is quite constructive once one follows some intuitions inherited from the classical case. The fused transfer matrices are by definition the “shifted” quantum analogue of the classical spectral invariants. So, it is natural to look for them (and for their eigenvalues) to satisfy a “shifted” quantum analogous of the spectral curve equation. Then, we are left just with the determination of the shifts in the argument of these transfer matrices and the computation of the corresponding coefficients in the TQ equations. These are indeed completely fixed by the known fusion relations and asymptotic behavior of the transfer matrices together with their known central zeroes. The second referee also referred to Algebraic Bethe Ansatz (ABA) to have a constructive derivation of the TQ equation. Indeed, this is quite natural in the rank one case along the lines we just described. Moreover, Sklyanin has shown that the Nested ABA form of the eigenvalues of the fused transfer matrices of the rank 2 case can be also used to reconstruct the TQ equations. However, one should mention that ABA by itself gives only an ansatz for the form of the transfer matrix eigenvalues and eigenvectors. The main problem in the ABA framework is to prove that indeed all the eigenvalues have the given form in terms of Q-functions, i.e. to address the completeness problem which is solved in the SoV framework.

Concerning the first Referee’s requests we have implemented them in the following way. For the request 1, we have modified the sentences containing the reference to ABA and we have added a footnote with the papers cited by the referee. For the request 2, we have changed some sentences referring to $Y_n$ to make more clear that they are the operators zeros of the commuting family of operators $B(\lambda)$. All the other requests have been implemented with the exception of 10 and 16.
Concerning 10 our x coincides with [n/2] for n odd and [n/2]-1 for n even. Concerning 16, the correct one is an $h_a$. In fact, this $a$ refers to the index appearing in the equations (A.27)-(A.30).
Note that as pointed out between (A.26) and (A.27), $h_a$ in the original covector appearing in (A.25)-(A.26) is assumed to be an integer between 0 and n-m-1. So that in the first step covector, appearing in (A.27)-(A.28), $h_a$ is incremented by one unit and so, it is an integer between 1 and n-m. Now by definition the value of $N^{+}_{m}$ of the second step covectors, appearing in (A.42)-(A.43), is one unit smaller compared to the one of the first step covectors, which reads $\bar{N}_{m}^{+}+\delta _{n-m}^{h_{a}}$. This last value is computed using the definition of $N^{+}_{m}$, equation (A.30) and recalling that in the first step covectors the value of $h_{a}$ is one unit bigger compared to the value that it has in the original covectors. We have added a note to clarify this point after equation (A.43). Finally, we have defined the function $a(\lambda)$ in equation (4.5) as this was indeed missing.

Concerning the other requests from the second and the third referees we have implemented them all.

Sincerely yours,

J. M. Maillet
G. Niccoli

List of changes

Both the referee’s 1 and 2 have asked about the homogeneous (and thermodynamic) limits. We have added a conclusion section to our paper, in particular to discuss this issue and also to give further comments on future developments. Here, let us just say that the general idea is to develop our analysis for both the spectrum and the dynamics in the inhomogeneous models and then to consider their homogeneous limit. This strategy has proven to be the most efficient in our previous works, in particular in the algebraic Bethe ansatz approach to correlation functions of quantum integrable lattice models, and we think this is also the best one in the present context. Indeed, considering first the inhomogeneous models enables one in general to give a simpler algebraic description of all the intermediate steps that would otherwise involve various (high) derivatives of the elementary objects involved. Moreover, let us remark that the characterization of the spectrum by TQ finite difference functional equation admits in general a smooth homogeneous limit. So, starting from the inhomogeneous complete characterization of the spectrum one gets the homogeneous one. However, one still has to prove that in the homogeneous limit this characterization is complete, i.e. that the full spectrum is described by the TQ equation. One way to prove it can be by direct construction of the SoV basis in the homogeneous models. This requires the proof of the non-degeneracy of the spectrum of the full set of fused transfer matrices in this limit. The other main subject which it is under analysis is the computation of scalar products and matrix elements on separate states for higher rank case. In the SoV framework for integrable quantum models related to gl(2), i.e. to rank one case, this type of analysis has been already developed with our collaborators N. Kitanine and V. Terras. Hence, the current analysis will represent a natural but non trivial extension of it.

The second referee asked also about a constructive derivation of the TQ equation in the SoV framework. We have added some comments about this in the Remark 4.1 after the Theorem 4.1 where this TQ equation is given. Let us say that from our point of view the way the TQ equation has been derived in our SoV framework is quite constructive once one follows some intuitions inherited from the classical case. The fused transfer matrices are by definition the “shifted” quantum analogue of the classical spectral invariants. So, it is natural to look for them (and for their eigenvalues) to satisfy a “shifted” quantum analogous of the spectral curve equation. Then, we are left just with the determination of the shifts in the argument of these transfer matrices and the computation of the corresponding coefficients in the TQ equations. These are indeed completely fixed by the known fusion relations and asymptotic behavior of the transfer matrices together with their known central zeroes. The second referee also referred to Algebraic Bethe Ansatz (ABA) to have a constructive derivation of the TQ equation. Indeed, this is quite natural in the rank one case along the lines we just described. Moreover, Sklyanin has shown that the Nested ABA form of the eigenvalues of the fused transfer matrices of the rank 2 case can be also used to reconstruct the TQ equations. However, one should mention that ABA by itself gives only an ansatz for the form of the transfer matrix eigenvalues and eigenvectors. The main problem in the ABA framework is to prove that indeed all the eigenvalues have the given form in terms of Q-functions, i.e. to address the completeness problem which is solved in the SoV framework.

Concerning the first Referee’s requests we have implemented them in the following way. For the request 1, we have modified the sentences containing the reference to ABA and we have added a footnote with the papers cited by the referee. For the request 2, we have changed some sentences referring to $Y_n$ to make more clear that they are the operators zeros of the commuting family of operators $B(\lambda)$. All the other requests have been implemented with the exception of 10 and 16.
Concerning 10 our x coincides with [n/2] for n odd and [n/2]-1 for n even. Concerning 16, the correct one is an $h_a$. In fact, this $a$ refers to the index appearing in the equations (A.27)-(A.30).
Note that as pointed out between (A.26) and (A.27), $h_a$ in the original covector appearing in (A.25)-(A.26) is assumed to be an integer between 0 and n-m-1. So that in the first step covector, appearing in (A.27)-(A.28), $h_a$ is incremented by one unit and so, it is an integer between 1 and n-m. Now by definition the value of $N^{+}_{m}$ of the second step covectors, appearing in (A.42)-(A.43), is one unit smaller compared to the one of the first step covectors, which reads $\bar{N}_{m}^{+}+\delta _{n-m}^{h_{a}}$. This last value is computed using the definition of $N^{+}_{m}$, equation (A.30) and recalling that in the first step covectors the value of $h_{a}$ is one unit bigger compared to the value that it has in the original covectors. We have added a note to clarify this point after equation (A.43). Finally, we have defined the function $a(\lambda)$ in equation (4.5) as this was indeed missing.

Concerning the other requests from the second and the third referees we have implemented them all.


Reports on this Submission

Report 2 by Thiago Silva Tavares on 2019-5-31 Invited Report

Strengths

Among the merits of the paper, the authors provide very technical proofs to a series of bijections which indeed completely characterize the spectrum of the model at hand. The proof for the $gl_n$ case is rather lengthier than the simpler $gl_2$ case and shows how one could compute the algebraic structure of the Bethe algebra. It seems to me this is an important step to our understanding of integrability in quantum systems. Therefore I would like to recommend the manuscript for a publication in SciPost Physics.

Weaknesses

No Weaknesses

Report

I would like to thank the authors for their response. They have addressed the raised points in a very satisfactory manner. Nevertheless, I would like include some other minor modifications to improve the writing of the manuscript. The list is suggestive.

Requested changes

Introduction, page 3, paragraph 1, line 2:
variables that we recently developed -> variables that we developed recently

Introduction, page 3, paragraph 2, line 2:
by the use -> by using

Introduction, page 4, paragraph 1, line 8:
quantum model -> quantum models

Introduction, page 4, paragraph 3, line 1:
allows us -> allows

Introduction, page 4, paragraph 3, line 5:
made coinciding -> made to coincide

Introduction, page 4, paragraph 3, line 6:
applies for -> applies to

Introduction, page 4, paragraph 4, line 18:
satisfying with the transfer matrices the same $n$ order… -> satisfying along with the transfer matrices the same order $n$…

Introduction, page 5, paragraph 1, line 19:
characterization -> characterizations

Introduction, page 5, paragraph 3, line 8:
it was there pointed out that -> it was pointed out there that

Section 2, page 6, just before (2.2):
it is solution -> it is a solution

Section 2, page 6, just before (2.5):
I prefer to call (2.5) the fundamental relation or just the Yang-Baxter algebra instead of calling it Yang-Baxter equation.

Section 2, page 7, just before (2.7):
… they are in generic position -> … they are in generic positions

Section 3, page 9, equation (3.9):
Maybe it would be better to change the name of the set $\{x_1,\ldots,x_N\}$. Maybe just $\Sigma$ without $T$.

Section 4, page 11, theorem 4.1:
Then an entire functions -> Then an entire function

Section 4, page 12, just before equation (4.9):
satisfies -> satisfy

Section 4, page 13, equation (4.20):
In equation (4.20) I suggest to let only $\varphi$'s on the RHS, in such a way that either you obtain the separate basis in terms of the transfer matrix with Lax operators normalized to be unitary (LHS), or by a specific combination of $Q$ operators (RHS).

Section 4, page 14, Corollary 4.1:
… let us denote with $k_j$ the corresponding eigenvalues -> let us denote $k_j$ the corresponding eigenvalues

Conclusion, page 18, paragraph 1, line 2:
Resolution -> solution

Conclusion, page 18, paragraph 2, line 13:
is true in this homogeneous limit -> is true in the homogeneous limit.


Conclusion, page 18, paragraph 2, line 14:
associated the -> associated to the

Conclusion, page 18, paragraph 2, line 21:
Is the fundamental first step -> is the first fundamental step

Appendix A, page 19, equation (A.12):
In this series of equations there is an index a with a different format

Appendix A, page 20, just before equation (A.17):
… in the case both -> in the case where both

Appendix A, page 21, just before equations (A.24)
N-tupla -> N-tuple

Appendix A, page 22, equations (A.36) and (A.41)
I believe that there is a misprint in the index of $N^+$. Instead of $m-1$ we should have $m+1$.

Appendix A, page 22, just after equation (A.48)
I believe that there is a misprint in the indices defining the loop

Appendix A, page 22, footnote, line 2:
Smaller of one unit of the one in the first step covectors -> one unit smaller than in the first step convectors

  • validity: top
  • significance: top
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2019-5-30 Invited Report

Report

The authors have answered to the questions I raised in a positive way
Their manuscript can be published

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  • originality: -
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