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All regular $4 \times 4$ solutions of the Yang-Baxter equation

by Luke Corcoran, Marius de Leeuw

This is not the latest submitted version.

Submission summary

Authors (as registered SciPost users): Luke Corcoran
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-10-25 12:17
Submitted by: Corcoran, Luke
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approaches: Theoretical, Computational


We complete the classification of $4\times 4$ regular solutions of the Yang-Baxter equation. Apart from previously known models, we find four new models of non-difference form. All the new models give rise to Hamiltonians and transfer matrices that have a non-trivial Jordan block structure. One model corresponds to a non-diagonalisable integrable deformation of the XXX spin chain.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-12-18 (Invited Report)


1. No substantial implications for physics.
2. Not mathematically clear enough as partially explained in the report.
3. Not sufficiently put in the research context.


The authors propose some new 4-by-4 matrix solutions of the
Yang-Baxter equation. These are listed in Section 4.1 of
their manuscript together with the corresponding
`local Hamiltonians'. From the exposition it is not clear if
these models have any applications in physics nor if
the fact that the R-matrices satisfies Equation (2.1) will
be of any true use in order, say, to diagonalize the
Hamiltonians or the associated transfer matrices.

Although I can imagine that the steps described in the
manuscript produce valid results, this has not become fully
clear to me from the exposition. Eq. (2.1) is not the
standard form of the Yang-Baxter equation (see e.g. the
English version of Wikipedia). The second and the third
factor on the left hand side as well as the first and the
second factor on the right hand side usually appear in
opposite order. In the Sutherland equations (3.11), (3.12)
the arguments of the R-matrices are suppressed. In this
form I cannot see if and how they follow from (2.1). It is
also not explained why these equations together with (2.2)
and (2.5) should determine a unique solution of (2.1).
The issue of the twist (see footnote 5) is not properly
discussed, also not in the cited work [11], which does not
seem to consider the most general case either. Moreover,
there are known 4-by-4 matrix solutions of the Yang-Baxter
equation which violate (2.2). Thus, even if the paper would
properly classify all joint solutions of (2.1)
and (2.2), which I doubt, the title would not be justified.

I also find it unpleasant that the authors do not put their
work in the context of the considerable and substantial work
that has been accumulated on the subject of the Yang-Baxter
equation. In particular, C. N. Yang and R. J. Baxter are
two living scientist who found the physically most relevant
4-by-4 matrix solutions of the Yang-Baxter equation and
understood part of the meaning and of the importance
of this equation. Their works are not cited. The original work
on the boost operator and on the Sutherland equation is not cited,
no connection is made with the theory of quantum groups and their
representations, with algebro-geometric approaches to solve the Yang-
Baxter equation or to Drinfeld's set-theoretic form of the equation.

Altogether I would say that the paper is not providing
enough new physical insight, its exposition is not mathematically
clear enough and the context is not explained well enough to
justify its publication in Scipost Physics.

  • validity: ok
  • significance: low
  • originality: ok
  • clarity: ok
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2023-12-5 (Invited Report)


1. A classification that was lacking, with new R-matrices unknown up to now (as far as I know)
2. Very clear


1. A bit repetitive w.r.t. the previous articles fo the authors (with collaborators)


The authors perform the classification of regular $4\times 4$ $R$-matrices with spectral parameters, completing the study they started with collaborators in two previous papers, [9] and [11]. The present paper deals with non-Hermitian $R$-matrices of non-difference form (meaning the $R$-matrix depends separately on the two spectral parameters). Non-Hermitian Hamiltonian are used in statistical physics, to study non-equilibrium models, justifying the need for such a classification.

The paper is well-written, and provides interesting results, including new $R$-matrices. I think it deserves publication, and I have just few minor points to correct, that I list below.

Requested changes

1. The paragraph around eqs (2.6)--(2.10) is a bit erratic: before (2.6) they start to explain the equation, but the full explanation comes at the end with eq. (2.10) and the sentence before: I would put directly the full explanation around eq (2.6), specially because (2.10) is in the middle of a discussion on conserved charges.

2. They should cite the reference [12] around eq (2.12), to justify this relation.

3. I don't understand the footnote 2: what means "not relevant"?

4. I don't understand the discussion after eq (3.15): it is known that the telescopic terms come from a gauge transformation of the $R$-matrix, eqs (3.1), (3.2). Then, since they classify the $R$-matrices up to these transformations, why the telescopic terms should play a non-trivial role in $Q_3$?

5. In section 4, they present the $R$-matrices they obtained. I suppose they checked that all their $R$-matrices obey the Yang--Baxter relation?
It is needed, since they work with necessary conditions only. As I said, I suppose they did it, but it is not mentioned (or I missed that point): it would be better to mention it explicitly.

6. Before section 4.1, they mention that they normalize the $R$-matrix in such a way that the (1,1) component is 1. They should add that it is always possible because $R$ is supposed to be regular, so that the (1,1) component cannot be 0.

7. In section 4.1, the denomination of the different models is not very clear. For the two first models, I suppose that "nilpotent" refers to the Hamiltonian, but I don't understand the name "trigonometric": does it refer to a deformation of the XXZ model, in the same way they use "deformed XXX" for the model 4? If yes, maybe "deformed XXZ" and "deformed XXX" for models 3 and 4 would be less confusing?

8. I think the section 4.3 is not needed, it is just a rephrasing of what they already did in their previous paper(s), and it adds only confusion w.r.t. section 4.2, which provides new informations.

9. Instead of their section 4.3, a comparison with the classification of constant $R$-matrices [20] would be more interesting. In particular, using the matrices of [20], can they perform a Baxterisation procedure to build all their spectral parameter dependent $R$-matrices?
That would give some insight on the algebraic structure which underlies the models they introduce, and I don't think it is a difficult check.

10. In section 4.4, they deal with Jordan decomposition in the case $L=4$. Do they have something to say on the general decomposition when $L$ is generic? I understand that a proof of the decomposition for $L$ generic is beyond the scope of the article, but they may have some educated guess based on computations for different values of $L$?

11. Finally, for three-state models, there are two classes of Hamiltonians related to elliptic curves, see I was not able to see the counter part of these Hamiltonians (and R-matrices) in the case of the two-state case dealt in the present paper. Could the author comment on this point?

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

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