# Algebraic Bethe Ansatz for spinor R-matrices

### Submission summary

 As Contributors: Vidas Regelskis Preprint link: scipost_202108_00062v1 Date submitted: 2021-08-25 11:21 Submitted by: Regelskis, Vidas Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Approach: Theoretical

### Abstract

We present a supermatrix realisation of $q$-deformed spinor-spinor and spinor-vector $R$-matrices. These $R$-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

#### Published as SciPost Phys. 12, 067 (2022)

Resubmission scipost_202108_00062v2 on 1 November 2021

You are currently on this page

Submission scipost_202108_00062v1 on 25 August 2021

## Reports on this Submission

### Anonymous Report 3 on 2021-10-25 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202108_00062v1, delivered 2021-10-25, doi: 10.21468/SciPost.Report.3731

### Strengths

1- New construction of R-matrices (of type spinor-vector and spinor-spinor) for the quantum algebras $U_q(\widehat{so_N})$ using the formalism of super-matrices.
2- Calculation of the the transfer matrices of closed spin-chains based on these algebras
3- Explicit diagonalization by the algebraic Bethe ansatz (which is new)
4- Clearly written

### Weaknesses

1- Rather technical

### Report

The paper is interesting and clearly written, despite its technicality.
I think it is worth publishing

### Requested changes

1- I find surprising that the $R$-matrices corresponding to $N$ even have a deformation parameter $q^2$, while the ones corresponding to $N$ odd have a deformation parameter $q$. There should be an explanation for this point.
In particular, it seems to me that removing the index 0 from the construction for $so(2n+2)$,
one should get the construction for $so(2n+1)$. Then, how the change in the deformation parameter occurs?
2-It would be nice to present also a super-matrix construction of the vector-vector $R$-matrix. In particular, the fusion of spinor-vector $R$-matrices should go back to the vector-vector $R$-matrix in a super-matrix form.
3-Since the presentation is quite technical, it is better not to use the same notation for different objects.
-- It is the case for the transposition $w$, defined in eq. 2.3 and differently on eq. 2.6. Then, in eq. 2.55, one does not know which transposition is used. On the transposition, the one introduced eq. 2.30 should be related to the ones already introduced.
-- In the same way, the operators in eq. 3.9 have nothing to do with the function $\beta$ used in eq. 2.36. A variation on the letter b (e.g. the mathfrak or mathbb styles, or any other style) would be better.
-- The notation used in eq. following line 138 is ambiguous: from what I understand $x_j^{m}$ is not $x_j$ to some power, but rather $x_j^{(m)}$ (which would be more consistent with the notation used elsewhere).
4-In the same spirit, some points should be precised to ease the reading of the manuscript:
-- When writing products of non-commuting operators, it should be indicated that the product is from left to right starting from the bottom index to the upper one in the product sign. This is particularly important for instance in eq. 3.8 (and should be reminded here), but it should be already stated in eq. 3.2
-- If I understand correctly, the notation $(\dot{a}\ddot{a})_n^j$ in eq. 3.10 means $\dot{a}_n^j\,\ddot{a}_n^j$. It should be stated explicitly.
-- I think also that it would be fair to indicate in the title that the article deals with $U_q(\widehat{so_N})$ algebras, but I leave it to the author.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Vidas Regelskis  on 2021-11-01  [id 1900]

(in reply to Report 3 on 2021-10-25)

I thank the referee for the points they have raised. I have provided an answer in the attached pdf.

### Anonymous Report 2 on 2021-10-12 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202108_00062v1, delivered 2021-10-12, doi: 10.21468/SciPost.Report.3654

### Strengths

1- Gives a new formula for the Bethe vectors of such models

2- All calculations are well described

### Report

I do recommend the publication of the paper.

### Requested changes

The author should precise the way he move the twist matrix trough the R matrix in the proof of Theorem 3.3 (from 3.22 and 3.23 to the nested transfert matrix in the next equation) and in the proof of Theorem 4.4 (from 4.29 and 4.30 to the nested transfert matrices in the next equation).

• validity: high
• significance: high
• originality: good
• clarity: high
• formatting: -
• grammar: -

### Author:  Vidas Regelskis  on 2021-11-01  [id 1899]

(in reply to Report 2 on 2021-10-12)
Category:
answer to question

I thank the referee for raising the point about way the twist matrices are moved through the R-matrices in the proof of Theorem 3.3 and Theorem 4.4. This was indeed overlooked in the first version paper. This issue has been fixed. The twist matrices are now included in the definition of the monodromy matrices, given by eqs. (3.2) and (4.3). (Tables on pages 19 and 26 were updated accordingly.) This is a valid construction since the twist matrices satisfy the fundamental exchange relations, (2.65) and (2.70). This yields the wanted results without needing to deal with the twist matrices explicitly.

### Anonymous Report 1 on 2021-9-5 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202108_00062v1, delivered 2021-09-05, doi: 10.21468/SciPost.Report.3497

### Report

I think that the paper "Algebraic Bethe Ansatz for spinor R-matrices" meets the requirements of the SciPost Physics Journal.

### Attachment

• validity: high
• significance: good
• originality: high
• clarity: good
• formatting: good
• grammar: good

### Author:  Vidas Regelskis  on 2021-11-01  [id 1898]

(in reply to Report 1 on 2021-09-05)
Category:
answer to question

I thank the referee for the point they have raised. I have provided an answer the attached pdf.