# Conjectures on Hidden Onsager Algebra Symmetries in Interacting Quantum Lattice Models

### Submission summary

 As Contributors: Yuan Miao Arxiv Link: https://arxiv.org/abs/2103.14569v3 (pdf) Date submitted: 2021-06-29 11:36 Submitted by: Miao, Yuan Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Quantum Algebra Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We conjecture the existence of hidden Onsager algebra symmetries in two interacting quantum integrable lattice models, i.e. spin-1/2 XXZ model and spin-1 Zamolodchikov-Fateev model at arbitrary root of unity values of the anisotropy. The conjectures relate the Onsager generators to the conserved charges obtained from semi-cyclic transfer matrices. The conjectures are motivated by two examples which are spin-1/2 XX model and spin-1 U(1)-invariant clock model. A novel construction of the semi-cyclic transfer matrices of spin-1 Zamolodchikov-Fateev model at arbitrary root of unity value of the anisotropy is carried out via transfer matrix fusion procedure.

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

I am grateful to the Referees for their assessment of our manuscript and for their valuable and detailed comments and suggestions. My replies to the queries and list of changes are given below.

1. I have added the proof of the Onsager algebra symmetries for XX model and spin-1 $U(1)$ invariant clock model using boost operator approach. The explicit expressions and recursive formulae for the higher-order Z/Y charges are derived and presented in the new appendices.

2. As the referee suggested, I show the isomorphism between the two algebras which are now called algebra $O$ and $O^\prime$ in the draft, see Appendix A. This provides the justification of denoting algebra $O^\prime$ as the Onsager algebra, which has been used already in Ref. [16] but not explicitly shown.

3. As the referee suggested, I add the discussion about the relation between the Onsager algebra and the $\mathfrak{sl}_2$ loop algebra in Appendix B. Specifically, I clarify the difference between the previously proposed the $\mathfrak{sl}_2$ loop algebra symmetries of XXZ model at root of unity and the conjectured Onsager algebra symmetries of XXZ model at root of unity in this draft.

4. As the referee pointed out, the spin-1 model with anisotropy parameter $\eta = \mathrm{i} \pi /3$ can be mapped into the same model with anisotropy parameter $\eta = 2 \mathrm{i} \pi /3$ via a unitary transformation. This is stated clearly in the first paragraph of Section 6 now.

5. I add the definition of $\mathbf{T}_s (u, \phi)$ and I move the commensurate condition for the twist $\phi$ before defining the Z/Y charges. I also complement Appendix C with the representations of $\mathcal{U}_q (\mathfrak{sl}_2)$ that have been used in the draft.

6. I change the typos and grammatical mistakes in the main text and bibliography. The references that have been suggested by the referees are added too, which I am grateful for the referees to point out this.

I would like to thank the referees again for the great suggestions. Should there be any more improvement, I would be appreciated to receive the comments.

### List of changes

1. I have added the definition of the transfer matrix $\mathbf{T}_s$ in Sec. 2.

2. I have added new proofs on the Onsager symmetries for XX model and spin-1 $U(1)$-invariant clock model using boost operator approaches. These are included in Sec. 4 and 6. A short review on the boost operator approach is added in App. F.

3. I have added the proof of the isomorphism of the two Onsager algebras mentioned in the draft, $O$ and $O^\prime$ in App. A.

4. A short discussion between Onsager algebra and $\mathfrak{sl}_2$ loop algebra is added in App. B.

5. Additional references that are relevant to the draft are included.

6. Several typos and grammatical errors are fixed.

### Submission & Refereeing History

Resubmission 2103.14569v4 on 19 July 2021

Resubmission 2103.14569v3 on 29 June 2021
Submission 2103.14569v2 on 21 April 2021

## Reports on this Submission

### Report

I thank the author for having improved the manuscript, which I now consider ready for publication in SciPost.

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### Report

The author has implemented changes required in my previous report. I recommend the paper to be published after correcting some typos.

### Requested changes

1. Correct a typo in the eq. 6.11

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### Strengths

Conjectured Onsager algebra symmetry for the quasi-periodic XXZ spin chain at roots of unity

### Report

We thank the author for the corrections.
The draft still needs to be improved, some important points remain to be clarified. Suggestions for changes are given below.

1. In the begining of section 4.2, the author implicitly identifies the Q's with the A's. That is OK for the XX model studied in section 4.1, as shown by (4.7)-(4.10) which supports the identification (4.12) and the use of the terminology 'Onsager generators' for the Q's'. However, that is not clear for the XXZ model at root of unity. For the Q's from (4.29)-(4.30), one should check that the relations of O' in order to claim the Q's as 'Onsager generators'. So, Conjecture I should be improved. It has two parts. One is the observation (4.28) holds. That's OK, numerical evidences are supporting this, as explained below (4.33).
But the second part of the conjecture is that (4.12) extends to the XXZ case. However, it is not clear if the author has checked that (4.29)-(4.30) satisfy the defining relations for O'. So, the following changes should be done:
- above (4.29), it should be said that (4.12) is conjectured.
- A reference to eq. (4.4) should be added above (4.31).
- Below conjecture I, a comment about the numerical check of (4.12) should be added.

2. Similarly to point 1, in section 6.2 a comment on the numerical check of (4.12) should be added.

3. Strictly speaking, in the Ising model the Onsager generators A0,A1 map to combinations of Pauli matrices according to (2.7). Clearly, the r.h.s of (2.7) generates a quotient of O: besides the defining relations for O, additional relations arise (shown by Davies years later). From that point of view, (2.7) should be improved by replacing equalities by --> (\rightarrow). Similarly, in the text the Q's are images of the Onsager generators in some quotient of the Onsager algebra. So, the Q's satisfy additionnal relations besides the ones for O or O'. In the text, it would be better to put '\rightarrow' instead of equalities when appropriate. For (4.12), it would be better to write

$A^r_m \rightarrow \frac{\ell_2}{4} Q^r_m$.

4. This comment may not be seen as important if the reader is a physicist, but might be for a mathematician. And because O' is new, it may be relevant for future works on the subject. The Onsager algebra has various presentations. The standard presentation denoted O, and the new one denoted O' by the author. So in the text, instead of saying 'the Onsager algebra O' and the 'Onsager algebra O'', it is better to specify at the begining of section 2 that two presentations of the Onsager algebra are used: the standard one denoted O and the other denoted O'. Then, in the Appendix although O and O' are a priori distinct at the begining, the proof shows that they are actually two presentations of the same object. Then, it is better to conclude that O and O' are two presentations of the Onsager algebra.

5. The wording should be double checked again. For instance: below (3.6), 'relaiton' --> 'relation'; below (4.19), 'Combing' --> 'Combining'.

6. The typo in ref. [36] 'ofUq(sl2)in' --> 'of $U_q(sl_2)$ shlould be corrected.

7. In [51], typo $u_q(sl_2)$ --> $U_q(sl_2)$

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