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

We would like to thank the referees for their detailed reviews. We have taken the feedback into consideration and made substantial changes to the submitted article. We reply to the comments of the reviewers separately.

Reviewer 1: We thank the reviewer for the feedback on our paper. We address the 11 points of this reviewer:

1: We agree that the discussion around (2.6)-(2.10) was a bit erratic. We modified this part of the text: we started with a definition of the transfer matrix, and described how the charges are obtained from this. We believe it is more logically sound and flows better this way.

2: We added the appropriate citation after (2.11).

3: Twists are model dependent, so we cannot use them to refine our initial ansatz for integrable Hamiltonians. We checked that none of the new models we found admit a twist which maps it into another one of our models. We have updated the footnote to give more details.

4: The logic is we start with a solution R of the Yang-Baxter equation and define the Hamiltonian density as H= P d_u R. While terms proportional to h2, h10, h12 do not appear in the integrable model Q2, they do appear in Q3 from the boost construction and are necessary to ensure [Q2,Q3]=0. By setting these to 0 from the beginning we would miss some models. We can indeed set h2=0 using a gauge transformation (3.2). However, we cannot set the others to 0 without interfering with the other gauge transformations we use, this is discussed in section 3.3.

5: We verified the Yang-Baxter equation for all of the models. We mention this now in the paper at the start of section 4.

6: We added that the (1,1) component of R is nonzero due to regularity.

7: The name 'trigonometric' is simply because of the appearance of a trigonometric function of u in the R-matrix, it has nothing to do with the trigonometric R matrix corresponding to the XXZ model. We removed the denomination 'trigonometric' due to this potential confusion.

8: We wanted to include all regular solutions to the Yang-Baxter equation in this paper for completeness and for reference. We believe it is useful to present all of these models in a uniform notation. We have clearly marked that section 4.1 is the one with new information, and that sections 4.2 and 4.3 are for completeness.

9: Baxterisation is a way to generate difference form solutions of the YBE from constant solutions. We consider non-difference form solutions, so our situation is slightly different. For example, our model 1 is given by R = P + A, where A is a nilpotent matrix. Baxterisation is usually applied to the case where A is invertible, see for example section 4.1 of 1310.5545. While trying to identify possible TL or Hecke algebra interpretations of our results is interesting, it's beyond the scope of our current paper.

10: We don't have any guesses for the Jordan block spectra at higher lengths. Computing these spectra symbolically is very computationally expensive, and the size of our Hamiltonians grow exponentially with the length of the spin chain. We can compute the spectra up to length 6, but higher than that a cleverer approach is needed. In this section we just wanted to highlight that these matrices are not diagonalisable and give a flavour or their spectra.

11: We didn't find any new elliptic models. There are, however, models containing the Jacobi elliptic functions cn and dn, and as such can related to an elliptic curve. These are mentioned in section 4.3.

Reviewer 2: We thank the reviewer for the feedback, which we have used to improve our paper. However, we strongly reject any insinuation that our results are incomplete. We have carefully carried out our analysis and double checked our results. We stand by our claim that we have found all regular 4x4 solutions of the YBE. While the physical applications for our new models are unclear at the moment, it is likely that they will find use given the recent interest in non-diagonalisable models.

There was an unfortunate typo in the Yang-Baxter equation (2.1), as the reviewer correctly pointed out. Indeed, the actual form of the Yang-Baxter equation we solve is R12(u,v)R13(u,w)23(v,w)=R23(v,w)R13(u,w)R12(u,v). The Sutherland equations we use in our paper follow from this form of the Yang-Baxter equation. We checked that all of our R-matrices satisfy this Yang-Baxter equation. We added some details to the derivation of the Sutherland equations from the Yang-Baxter equation, without suppressing any arguments. We hope that this is more clear mathematically.

The Sutherland equations (3.13) and (3.14) in the latest draft and the commutation [Q2,Q3]=0 are necessary conditions for the YBE (2.1) to hold. We first classify all potentially integrable Hamiltonian densities H by solving [Q2,Q3]=0, where Q3 is constructed from H using the boost operator. This task is simplified by first making use of gauge transformations (basis transformations, scaling, reparametrisation) at the level of the R-matrix, allowing us to restrict the entries of H. We do not include twists in our initial gauge transformations because these are model dependent, so they cannot be used on a general solution R(u,v) to simplify our ansatz for H. The only potential danger is members of our final list of integrable Hamiltonians being related by a twist transformation, which we found not to be the case.

In response to 'It is also not explained why these equations together with (2.2) and (2.5) should determine a unique solution of (2.1).' As mentioned, the Sutherland equations are a priori only necessary conditions for the YBE to hold. For each integrable Hamiltonian H we found a unique solution to this pair of first order differential equations for the entries of R(u,v), after supplying the pair of boundary conditions (2.2) and (2.5). The Sutherland equations have not been proven to be a sufficient condition for the YBE to hold; as stated in the paper this is still a conjecture. Therefore we check afterwards that each solution to the Sutherland equations (3.13) and (3.14) is indeed a solution of the YBE.

In response to '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'. The title of the paper is 'All regular 4x4 solutions of the Yang-Baxter equation'. Regularity is exactly equation (2.2), so our title is valid. We know that there are many interesting non-regular solutions (see for example the recent paper 2401.12710), but that is beyond the scope of this work and what is possible with the boost operator.

It is not yet clear how use the Yang--Baxter equation to diagonalise the associated transfer matrices. Indeed this an important question for future research. Into this direction, there is a recent paper (2309.10044) which classifies the Jordan block structure of an non-diagonalisable integrable model using the symmetries of the R-matrix.

We have improved the historical references in the paper. We have cited the original papers of Yang and Baxter, as well as the first instances of the boost operator and the Sutherland equations. We have mentioned more algebraic approaches to solving the YBE, including Baxterisation.

1: Improved historical references, cited original works of Yang/Baxter/Sutherland/Tetel'man.

2: Improved introduction - added more references on algebraic approaches to solving YBE, e.g. Baxterisation.

3: Fixed typo in (1.1) and (2.1), gave more detail on derivation of Sutherland equations.

4: Reshuffled discussion around (2.6)-(2.10) to make the logic flow better.

5: Added appropriate citation after (2.11)

6: Improved discussion on twist, upgraded footnote to short paragraph. Twists are not relevant for our approach, we just need to check that in the end that our new models are not related by them.

7: Added explicitly that we verify the YBE for all new models, added this to the "method" section.

8: Added that we can normalise the (1,1) component of R to 1 using regularity.

9: Removed trigonometric denomination of model 3 due to potential conflation with XXZ model.