A range three elliptic deformation of the Hubbard model

We are thankful to the referees for the truly thorough review of our manuscript. We have made many changes to the
text. Most of the changes are minor, but there are a few bigger changes. The most significant change is that we rewrote
Section 5.2, because we also found that it was not clear enough. Below we detail the many changes that we made.

Reply to Ref. 1

We changed the structure of section 5 and followed the suggestions.
1. We added the definition of $\check{R}$ . We also added that the commutation relations between L comes from [R12,R34]=0
2. We have removed the function G, all the derivation should now be more straightforward.

MINOR POINTS
1. Added a comment after (2.27)
2. Changed sentence as referee suggested.
3. Added clarification.
4. Yes it includes mixture of operators based on sigma and tau. Added footnote 2 at page 7 to clarify.
5. Regarding the sector with odd down spins: Here we did not add any new explanations. This would be a very long
discussion. In any case, the situation is that we do not know of any examples, where the oddity of the number of spins
would influence a thermodynamic quantity, in the bulk, for a quantum spin chain. It is known that boundary conditions
can have effect, but in other types of problems, where the boundary condition is more severe: for example the 6-vertex
partition function with periodic vs. fixed boundaries. But that is a very different problem.
6. To clarify, we added at the beginning of the sentence "First we transform the kinetic terms (given by (2.16))" and we
also added as suggested "and redefining j->j+1/2".
7. Checked and corrected.
8. We added a new sentence about this: the family H2 does not include the usual Hubbard model either.
9. Corrected.
10. Corrected.
11. Added footnote 8 about it.
12. Modified with Tr_ab.
13. Added R in term of Pauli Matrices.

Reply to Ref 2

1. Brackets added.
2. Corrected.
3. We added a new sentence summarizing the essence of the work [32].
4. Corrected.
5. Indeed, the transfer matrix construction will need the regularity of the Lax operator, corrected.
6. Added a comment after (5.9). The reparametrization (5.8) is general, but we chosed (5.9) for this model so that the
entries of the R matrix can be simplified by using some properties of the Jacobi functions.
7. We added a footnote to clarify, at the end of page 13.
8. Thank you, we have added this comment after (5.10) and also in the appendix after A.4.
9. We have rewritten that part to make the statements clearer.
10. We added a sentence about this at that place: yes, this is generic for models obtained via bond-site transformation.
11. We reformulated that part.
12. We reformulated that part.
13. Corrected.
14. We have added a better explanation at the beginning of Sec. 6.
15. We decided to add a Mathematica notebook.

Reply to Ref 3

1. Commutation relations added.
2. We exchanged the notation.
3. We added a sentence about this.
4. We added a footnote about this on page 7.
5. We rewrote Section 5.2.
6. We added the expression of the next conserved charge after Q3 in the appendix B. This is a range 5 operators. We
decided not to add other charges, since their expressions will be really long, but the interested reader can compute them
using the definition of the transfer matrix.
7. We added a footnote about this question, after the comment that H2 does not degenerate to the Hubbard model.
8. This is replaced.
9. Corrected.
10. Corrected.
11. Added a footnote at pag 2. and a comment after A1.
12. We think that the R-matrix (A.1) does not reduce to the standard Hubbard model in any limit and since [24] is a
deformation of the Hubbard model it does not reduce to it in any limit of κ.
13. Corrected, now we write "two-site" and "three-site" at every occurance.