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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
Preprint Link:  (pdf)
Date accepted: 2022-08-23
Date submitted: 2022-08-18 15:15
Submitted by: Miao, Yuan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

Published as SciPost Phys. 13, 070 (2022)

Author comments upon resubmission

Dear referee and editor,

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 text about whether the Fendley model with periodic boundary condition is interacting. The improved part in page 9 is
However, the spectra of the Hamiltonians do not satisfy the free fermionic condition in [53] with the periodic boundary condition. This can be observed from numerically obtaining the eigenvalues of (3.14), as shown in Fig. 2. In principle, this do not exclude the possibility that the spectra cannot be partitioned into subsectors that are free fermionic. The most notable example is the TFIM with periodic boundary condition, where the spectrum can be divided into two parts that are free fermionic. In the case of the Fendley model, it is less clear whether such partition of the spectrum into free fermionic parts exists. Moreover, unlike the TFIM, the non-local transformation constructed in [33] no longer applies for periodic boundary. The question whether the Fendley model is intrinsically interacting is postponed to future investigation.
Upon the comments of the referee, it is difficult to determine whether the model is interacting or the spectra could be partitioned into parts that are free fermionic. The referee commented that maybe one can divide the spectra into parts that are free. Indeed, it seems possible. But with different degeneracies, it is hard to construct a systematic way of doing so, in my opinion. This seems to be different from transverse field Ising model.

2. I agree with the referee's comment and I added the following remark in page 9:
Remark. The transfer matrix in [33] can be applied to the $r = 2$ Fendley model with inhomogeneous couplings and periodic boundary. However, the method in Sec. 4.2 only works for the homogenous case (3.16). Instead, we can add inhomogeneities in the transfer matrix (4.22) constructed in Sec. 4.2, which will result in another Hamiltonian with longer-range interaction.

3. The broken reference link in page 2 is fixed.

4. I corrected a typo in Eq. (B.2) and a wrong statement that Ref. [66] can only be applied to R matrices of difference form.

Reports on this Submission

Anonymous Report 2 on 2022-8-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2203.16594v4, delivered 2022-08-20, doi: 10.21468/SciPost.Report.5565


This is a well written, very useful paper where the authors clarify relations between various fancy algebras related with integrable models - Temperley-Lieb, Onsager - and further, point out relationships with recent models of interest such as the ones proposed by P. Fendley. In what is usually a rather technical and somewhat unnecessarily mathematical area, I found the paper refreshing, and I am sure it will become a standard reference.


There were points that deserved improvement, but these were taken take of thanks to the work of the first referee - sorry for contributing my own report a little late.


I think the paper is now ready for publication.

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

Anonymous Report 1 on 2022-8-20 (Invited Report)


I thank the author for clarifications and changes. In my view, the revised version of the paper is suitable for publication in SciPost.

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

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