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Generalised Onsager Algebra in Quantum Lattice Models
by Yuan Miao
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Submission summary
| Authors (as registered SciPost users): | Yuan Miao |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2203.16594v3 (pdf) |
| Date submitted: | 2022-08-05 11:05 |
| Submitted by: | Miao, Yuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley-Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physical properties related to the generalised Onsager algebras, hinting at a general framework that includes families of quantum lattice models possessing the structure of the generalised Onsager algebras.
Author comments upon resubmission
I am grateful for the referee for his valuable comments and suggestions on the draft. I have improved the draft according to the referee's suggestions. The list of changes are given below
List of changes
1. I have improved the part about the quotient of $\mathrm{GTL} (\sqrt{2}, r , N)$ around (2.5). The map (2.5) has been stated more clearly now.
2. Below (2.6) I have given the precise reference of the Definition in [1].
3. I have corrected the maps in (2.11) and (2.12) according to the referee's suggestion.
4. The part about (2.15) commutes with $\mathbf{A}_n$ has been rewritten.
5. I added a sentence to explain why we review the known cases in the first paragraph of Sec. 3.
6. I have improved the language used near (3.1), (3.2), (3.7) and Sec. 3.3 according to the referee's suggestion. I deleted the previous footnote near (3.7).
7. Below (3.6), I have explained the origin of the terminology "spin current" as the spin current in spin-1/2 XXZ model.
8. About 13. in the referee's report, I added a figure (Fig. 2 in the updated version) and explained the reason why Fendley models with periodic boundary should be interacting; namely, the non-local transformation in Fendley '19 does not apply any more. But Jordan-Wigner transformation (for TFIM) still works with periodic boundary. This has been observed in Fendley '19 already, and stated explicitly in its abstract.
9. About 14. in the referee's report, in fact in Fendley '19, the author constructed extensively-many conserved quantities for the Fendley model with open boundary condition and inhomogenous couplings. This will not work for the periodic boundary condition. In the periodic case, I used the Lax operator to obtain transfer matrix. In principle, I could put inhomogeneities in the Lax operator and get a model that is still integrable. However, it is not the same Fendley model with inhomogeneous couplings, but a model with local density of longer-range terms. That is not what Fendley '19 studied. In short, the statement about Fendley model with periodic boundary and inhomogeneous couplings is not integrable stays correct.
10. I changed the word "the generalisation of the phase transition in the TFIM" into "analogous to the phase transition in the TFIM".
11. About 22. in the referee's report, I have checked briefly and the R matrix (B.1) does not fit the known categories in the two references. I have made a short comment in the updated version.
12. I have added the references recommendated by the referee. Mainly in the introduction and the part related to the baxterisation.
13. I have changed typos mentioned in the referee's report (9., 11., 12., 15., 18., 19. 20., 23. and 24. there).
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 3) on 2022-8-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2203.16594v3, delivered 2022-08-15, doi: 10.21468/SciPost.Report.5538
Report
I thank the author for the revised version. Most of the points addressed in the previous report have been answered properly or corrected. However, three points remain to be completed or clarified, and a typo should be corrected.
a. Below (2.6), a comment about the possible existence of additionnal relations should be given. Is the quotient characterized only by (2.3),(2.6) or additionnal relations may appear? If the author doesn't know, that should be said explicitly.
b. About 8 of the first report. The answer is not satisfying in the
present revised form. Indeed, to support the hypothesis that Fendley's model for periodic boundary conditions (pbc) is not interacting (at least for small sizes), let us observe the following. It is correct to say that Fendley's method only provides the quasi-energies and raising/lowering operators for the open boundary case. However, this does not exclude the possibility of finding them using some other way (yet to be found) for the pbc case. In fact, for small lattices one can indeed find the quasi-energies by brute force. For the case depicted in Fig.2, let e_1=\sqrt{3},e_2=1+\sqrt{2},e_3=1-\sqrt{2}. One can find by brute force that the eigenvalues of Fig.2 are given by,
e_1+e_1
-e_1-e_1
e_1-e_1
e_2 + e_3
e_2 - e_3
- e_2 + e_3
- e_2 - e_3
This case may look trivial but larger chains could be analyzed by brute force as well. Note that the above structure is also present in periodic Ising. Indeed, the definition (1) in ref [53] does not really apply to the Ising case with pbc - this is related to the fermionic parity sectors. So, maybe Fendley's model with pbc is similar to Ising with pbc.
So, if the author asserts that Fendley's model for pbc is interacting, that should be clearly proven. I can not find such statement in Fendley's paper. So, the author should indicate the precise location in the text of Fendley's paper where such statement is justified, or give a precise reference in the literature, or give a proof of it (contradicting the above examples for small sizes). Otherwize, the claim that Fendley's model for pbc is interacting should be removed from the text. Saying that it is hard to solve it for pbc is a different issue.
c. About point 9 of the first report. The answer is not satisfying in the present revised form. Indeed, while the transfer matrix approach for the open case does not apply to periodic pbc, in Appendix A of ref. [33] a different approach is provided, and it is also suitable for inhomogeneous couplings (as explained in the last paragraph of Appendix A of [33]). So, whereas it is correct to say that the approach proposed by the author (different from [33]) exhibits integrability, it doesn't exclude the possibility that the model is integrable for inhomogeneous couplings. Being integrable or admitting a Lax operator construction are not equivalent statements. The second statement implies the first, but the reverse is not true in general.
d. Typo middle page 2 in introduction: the q-Onsager algebra [?, 24, 25]
Provided the above changes, I think the new revised version is suitable for publication in SciPost.
Requested changes
See report