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Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra
by Oleg Lychkovskiy
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Submission summary
Authors (as registered SciPost users):  Oleg Lychkovskiy 
Submission information  

Preprint Link:  https://arxiv.org/abs/2012.00388v4 (pdf) 
Date accepted:  20210517 
Date submitted:  20210427 09:58 
Submitted by:  Lychkovskiy, Oleg 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic manybody system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3state Potts model.
Author comments upon resubmission
The second Referee and the Editor have requested to comment upon the applicability of the method to a wider range of integrable models. I agree with the second Referee that most integrable models are not known to have a simple underlying algebraic structure similar to the Onsager algebra. A straightforward generalization of our method to such models is therefore unlikely. I would like to point out, however, to a very recent ref. [61], where a hidden Onsager algebra has been conjectured for the integrable XXZ spin1/2 chain at the rootofunity anisotropies. I have added these considerations to the "Summary and outlook" section. I believe further work could elucidate the scope of the method.
List of changes
1) Following the advise by the second Referee, I have added the following paragraph to the "Summary and outlook" section:
An interesting open question is whether the method presented here can be extended to a broader range of integrable model. Most integrable models are not known to possess a simple algebraic structure analogous to the Onsager algebra (see, however, a recent ref. [61] where a hidden Onsager algebra has been conjectured for the integrable XXZ spin1/2 chain at the rootofunity anisotropies). The absence of such structure prevents a straightforward generalization of the method.
2) I have mentioned that my result in a particular case agrees with a very recent preprint [56], see the text below eq. (23).
3) I have added a line below eq. (31) explaining how this equation depends on a_0.
4) Several references have been updated.
Published as SciPost Phys. 10, 124 (2021)
Reports on this Submission
Anonymous Report 2 on 202153 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.00388v4, delivered 20210503, doi: 10.21468/SciPost.Report.2870
Report
The author has added a remark on the possible extension to integrable models without an algebraic structure similar to the one considered in the manuscript and added a reference to a conjectured 'hidden' Onsager algebra structure in the XXZ spin chain at certain values of the anisotropy.
I recommend to publish the paper in the present version in SciPost Physics.
Anonymous Report 1 on 202152 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2012.00388v4, delivered 20210502, doi: 10.21468/SciPost.Report.2865
Report
The author's could also look to
T. Deguchi, K. Fabricius and B.M. McCoy, J. Stat. Phys. 102, 701 (2001)
about loop algebra symmetry at root's of unit.
Author: Oleg Lychkovskiy on 20210506 [id 1413]
(in reply to Report 1 on 20210502)I thank Referee for bringing this reference to my attention. It would be interesting to try to extend the approach developed in my paper by exploiting the said algebraic structure.