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On quantum separation of variables beyond fundamental representations

by J. M. Maillet, G. Niccoli

Submission summary

As Contributors: Jean Michel Maillet · Giuliano Niccoli
Arxiv Link: https://arxiv.org/abs/1903.06618v2 (pdf)
Date accepted: 2021-01-14
Date submitted: 2021-01-06 14:31
Submitted by: Maillet, Jean Michel
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Mathematical Physics
Approach: Theoretical

Abstract

We describe the extension, beyond fundamental representations of the Yang-Baxter algebra, of our new construction of separation of variables bases for quantum integrable lattice models. The key idea underlying our approach is to use the commuting conserved charges of the quantum integrable models to generate bases in which their spectral problem is separated, i.e. in which the wave functions are factorized in terms of specific solutions of a functional equation. For the so-called "non-fundamental" models we construct two different types of SoV bases. The first is given from the fundamental quantum Lax operator having isomorphic auxiliary and quantum spaces and that can be obtained by fusion of the original quantum Lax operator. The construction essentially follows the one we used previously for fundamental models and allows us to derive the simplicity and diagonalizability of the transfer matrix spectrum. Then, starting from the original quantum Lax operator and using the full tower of the fused transfer matrices, we introduce a second type of SoV bases for which the proof of the separation of the transfer matrix spectrum is naturally derived. We show that, under some special choice, this second type of SoV bases coincides with the one associated to Sklyanin's approach. Moreover, we derive the finite difference type (quantum spectral curve) functional equation and the set of its solutions defining the complete transfer matrix spectrum. This is explicitly implemented for the integrable quantum models associated to the higher spin representations of the general quasi-periodic Y(gl2) Yang-Baxter algebra. Our SoV approach also leads to the construction of a Q-operator in terms of the fused transfer matrices. Finally, we show that the Q-operator family can be equivalently used as the family of commuting conserved charges enabling to construct our SoV bases.

Published as SciPost Phys. 10, 026 (2021)



Author comments upon resubmission

Dear Editor,

We first would like to thank the referees for their careful reading of the manuscript, for their clarification requests and for pointing out numerous misprints and suggesting several improvements in the notations and also concerning the English grammar. We have implemented the corresponding modifications in our manuscript to take them into account, the mains changes being in Section 2 and in Section 4. We also added a new section, “Conclusion”, and an appendix on the determinant of tri-diagonal matrices that we use for obtaining new fusion relations that enable us to construct in two different ways our SoV bases.

To list the main modifications done in the manuscript (see below), we have answered separately the points raised by each of the two referees.

Best regards,

J. M. Maillet and G. Niccoli

List of the main modifications done in the manuscript:

Answer to Referee 2:

We agree with all remarks of the referee and we have implemented all his requirements.

  1. We have stated that P_{a,b} is the permutation operator.

  2. We have added the missing ^{h_n} in equation (4.3).

  3. As suggested, we have moved the former Proposition 4.2 in the current Proposition 2.3 of Section 2. We kept this proposition as we use it to prove our discrete characterization of the transfer matrix spectrum given in Theorem 4.1.

  4. In our new version of Theorem 4.2 we have presented the construction of the SoV basis both starting from the co-vector <S| (of maximal values of the coordinates h_i = 2 s_i, for each i) and from the co-vector <O| (with coordinates h_i=0 for all I’s), which coincides with the co-vector <0| used in the first Sklyanin’s construction in the case a) of this Theorem. This should clarify the different possible constructions of this SoV basis as required by the referee. The construction of this SoV basis starting from <O| has required the introduction of some additional set of fusion relations introduced in our new Proposition 2.4.

Finally, we have added a section “Conclusion” where we discuss on general grounds the relations between the different SoV bases introduced in the paper. We hope it will enable the reader to have a broader view of the possibilities opened by our new approach to SoV bases.

Answer to Referee 1:

We have implemented mainly all the suggestions and requests of the referee. Then, in the following, we only comment on the few that we didn’t implement completely, or which require some additional explanations.

About 13- We have changed the notation for the quantum determinant.

About 20- We have removed formula (2.30) and explained it by words after current Proposition 2.3 (note that we also enlarged that property to give all central zeroes of the higher fused transfer matrices).

About 24- We have added the missing -1 and introduced in (3.5) the missing definition of the \mathcal{W}.

About 26- We prefer to keep the notation {Sk} instead of using {\text{Sk}}.

About 33- We would like to keep (4.7) in the current form.

About 36- The difference between the D_{t,n} with or without the superscript is now explained in remark 4.

About 37- The current (4.8) (old (4.10)) is the discrete system of equations completely characterizing the spectrum of the transfer matrix in the SoV basis. We prove that it is equivalent to the characterization in terms of a Baxter’s type TQ-equation which in turn leads to Bethe equations. This is a link with the ABA, however, we would like to stress that we have not done any Ansatz in deriving them. About what is simpler to solve one should say that to our knowledge there do not exist general tools to solve general system of polynomial equations with degree higher than one. Then, one should look to the specific system that we are considering determining if some simplifications emerge (as it should be, due to integrability); this is an open and interesting question. Instead, we have been interested in the reformulation of these polynomial equations in terms of a functional equation (spectral curve equation) leading in its turn to Bethe equations; this is motivated by the fact that for Bethe equations the existing literature is more developed, in particular concerning the analysis of the thermodynamic limit.

About 39- We have defined what common component means and also restated more precisely the use of the Theorem of Bez’out.

About 43- We think that it is appropriate to keep our notation to properly implement the induction as we are proposing it. The N-tuple of \bar h is required to define the level up to which for all the h_i, bigger or equal of the corresponding \bar h_i, we assume that our statement is already satisfied. Now to be able to prove the induction, we have to show that we can move this level reducing of one unit each \bar h_n. To prove it we need that all the h_i for i different of n must be in generic value bigger or equal of the corresponding \bar h_i, we should not just prove it keeping the others h_i on the level \bar h_i. Concerning, the \hat h_n this has been used only to make explicit that we are fixing the value of the h_n to 2s_n.

About 54- We have reformulated the Corollary 5.2 to take into account the suggestion of the referee.

About 53- The condition to be satisfied by any proper “reference co-vector” starting from which an SoV basis can be constructed is the condition (5.46). Now, the “reference co-vector” <L| is written in terms of <S| by an invertible charge in (5.47), so the condition required on <L| is satisfied if <S| satisfies (5.46). In our new section “Conclusion”, we have added a related discussion. In particular, we have shown that a change of “reference co-vector” must always be implemented by the action of an invertible charge on a given “reference co-vector” which is known to satisfy (5.46). So that for the two SoV bases constructions, presented in this paper, this condition explicitly reads in (6.7) and in (6.8).

About 56- We would like to point out that the discrete spectrum characterization of the original transfer matrix, given in Theorem 4.1, defines a simultaneous characterization of the spectrum of all the fused transfer matrices and so also of the fundamental transfer matrix. Indeed, from the simplicity of the transfer matrix spectrum, for any solution of the system (4.8) we get that the formula (2.26) gives also the eigenvalues of the fused transfer matrices. This is done simply by replacing the original transfer matrix in (2.26) by its eigenvalue. The referee probably is asking about the possibility to derive directly a discrete system of equations for the fundamental transfer matrix spectrum. From our previous discussion this system must be equivalent to the one already derived in Theorem 4.1. It might be however interesting to derive it directly, and this should be done using the fusion relations starting from the fundamental transfer matrix and not from the original one.

List of changes

To list the main modifications done in the manuscript, let us answer separately the points raised by the two referees.

Answer to Referee 2:

We agree with all remarks of the referee and we have implemented all his requirements in the new version of our manuscript.

1. We have stated that P_{a,b} is the permutation operator.

2. We have added the missing ^{h_n} in equation (4.3).

3. As suggested, we have moved the former Proposition 4.2 in the current Proposition 2.3 of Section 2. We kept this proposition as we use it to prove our discrete characterization of the transfer matrix spectrum given in Theorem 4.1.

4. In our new version of Theorem 4.2 we have presented the construction of the SoV basis both starting from the co-vector <S| (of maximal values of the coordinates h_i = 2 s_i, for each i) and from the co-vector <O| (with coordinates h_i=0 for all i’s), which coincides with the co-vector <0| used in the first Sklyanin’s construction in the case a) of this Theorem. This should clarify the different possible constructions of this SoV basis as required by the referee. The construction of this SoV basis starting from <O| has required the introduction of some additional set of fusion relations introduced in our new Proposition 2.4.

Finally, we have added a section “Conclusion” where we discuss on general grounds the relations between the different SoV bases introduced in the paper. We hope it will also enable the reader to have a broader view of the possibilities opened by our new approach to SoV bases.

Answer to Referee 1:

We have implemented mainly all the suggestions and requests of the referee. Then, in the following, we only comment on the few that we didn’t implement completely, or which require some additional explanations.

About 13- We have changed the notation for the quantum determinant.

About 20- We have removed formula (2.30) and explained it by words after current Proposition 2.3 (note that we also enlarged that property to give all central zeroes of the higher fused transfer matrices).

About 24- We have added the missing -1 and introduced in (3.5) the missing definition of the \mathcal{W}.

About 26- We prefer to keep the notation _{Sk} instead of using _{\text{Sk}}.

About 33- We would like to keep (4.7) in the current form.

About 36- The difference between the D_{t,n} with or without the superscript is now explained in remark 4.

About 37- The current (4.8) (old (4.10)) is the discrete system of equations completely characterizing the spectrum of the transfer matrix in the SoV basis. We prove that it is equivalent to the characterization in terms of a Baxter’s type TQ-equation which in turn leads to Bethe equations. This is a link with the ABA, however, we would like to stress that we have not done any Ansatz in deriving them. About what is simpler to solve one should say that to our knowledge there do not exist general tools to solve general system of polynomial equations with degree higher than one. Then, one should look to the specific system that we are considering determining if some simplifications emerge (as it should be, due to integrability); this is an open and interesting question. Instead, we have been interested in the reformulation of these polynomial equations in terms of a functional equation (spectral curve equation) leading in its turn to Bethe equations; this is motivated by the fact that for Bethe equations the existing literature is more developed, in particular concerning the analysis of the thermodynamic limit.

About 39- We have defined what common component means and also restated more precisely the use of the Theorem of Bez’out.

About 43- We think that it is appropriate to keep our notation to properly implement the induction as we are proposing it. The N-tuple of \bar h is required to define the level up to which for all the h_i, bigger or equal of the corresponding \bar h_i, we assume that our statement is already satisfied. Now to be able to prove the induction, we have to show that we can move this level reducing of one unit each \bar h_n. To prove it we need that all the h_i for i different of n must be in generic value bigger or equal of the corresponding \bar h_i, we should not just prove it keeping the others h_i on the level \bar h_i. Concerning, the \hat h_n this has been used only to make explicit that we are fixing the value of the h_n to 2s_n.

About 54- We have reformulated the Corollary 5.2 to take into account the suggestion of the referee.

About 53- The condition to be satisfied by any proper “reference co-vector” starting from which an SoV basis can be constructed is the condition (5.46). Now, the “reference co-vector” <L| is written in terms of <S| by an invertible charge in (5.47), so the condition required on <L| is satisfied if <S| satisfies (5.46). In our new section “Conclusion”, we have added a related discussion. In particular, we have shown that a change of “reference co-vector” must always be implemented by the action of an invertible charge on a given “reference co-vector” which is known to satisfy (5.46). So that for the two SoV bases constructions, presented in this paper, this condition explicitly reads in (6.7) and in (6.8).

About 56- We would like to point out that the discrete spectrum characterization of the original transfer matrix, given in Theorem 4.1, defines a simultaneous characterization of the spectrum of all the fused transfer matrices and so also of the fundamental transfer matrix. Indeed, from the simplicity of the transfer matrix spectrum, for any solution of the system (4.8) we get that the formula (2.26) gives also the eigenvalues of the fused transfer matrices. This is done simply by replacing the original transfer matrix in (2.26) by its eigenvalue. The referee probably is asking about the possibility to derive directly a discrete system of equations for the fundamental transfer matrix spectrum. From our previous discussion this system must be equivalent to the one already derived in Theorem 4.1. It might be however interesting to derive it directly, and this should be done using the fusion relations starting from the fundamental transfer matrix and not from the original one.

Submission & Refereeing History

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Resubmission 1903.06618v2 on 6 January 2021

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