Lifting integrable models and long-range interactions

See the attached Requested_changes.pdf

1. innovative
2. rigorous

This paper addresses the question of whether one can construct a Lax operator and and R-matrix in order to demonstrate that a given spin-chain Hamiltonian is integrable. This is done not trying to match the Hamiltonian to one of the classes which are known to be integrable (by the classification of the same authors and collaborators), but directly embedding the Hamiltonian in the tower of charges and constructing the Lax operator and R-matrix algorithmically starting from small number of sites. The procedure is practically and most efficiently carried out by a computer programme. This method is then extended to lang-range spin-chains, for example of the type appearing in the AdS/CFT correspondence.

The paper is interesting and it brings forward previous work on this line of investigation, and it constitutes progress in determining the complete algebraic structure underlying the AdS_5/CFT_4 integrable system. I definitely recommend it for publication. I only have a few minor comments which the authors might perhaps find useful to consider:

1. It would be useful to recollect at the beginning what the difference between a "constant" and "functional" Hamiltonian is

2. The meaning of P_{ia} as the permutation operator is never stated

3. The connection with the SU(2) sector in N=4 super-Yang-Mills is advertised but never truly made very explicitly. It is not quite clear whether this is something which this method will allow to do in the future, or whether hidden in the formulas of this paper one can already find for instance the higher loop Lax operator and R-matrix for AdS/CFT. For one thing, there is no recap of the N=4 dilatation operator expression (perhaps with a little comment on the original notation of the N=4 literature confronted with the one of this paper) for the ease of comparison. Since this is such a central point of the method, it would be good to have a very clear presentation of this aspect in a completely clear and unequivocal way if possible.

Please see report above

1- The paper addresses an interesting technical question in the context of integrable spin chains. 2- It is mostly well written and provides a number of useful examples.

1- Some definitions could be sharpened or better explained. 2- The usefulness of the results could be addressed in more detail.

1- The paper refers to a "constant" Hamiltonian as opposed to a "functional" one. These to terms should be defined explicitly. 2- Below eq. (1.1) it is stated that these spin chains are integrable. This should probably mean perturbatively integrable, which should be briefly explained at this place to avoid confusion. 3- Above (2.19) it is stated that R is the unique solution to the Sutherland equation. Is this obvious? 4- The construction around (2.21)-(2.23) is a bit intransparent. The A in the equation below eq. (2.23) carries no site labels, while one term on the right hand side has these labels. How is that to be understood? Does this A enter into the boost operator in (2.22), which would require a density? If yes, what is the density of H^2 in the definition of A below eq. (2.23)? Please clarify these points. It would also be helpful to give an explicit example for Q3 in eq. (2.23), in particular for the contribution H^(1)_{k,k+1}. 5- Is there a reason to use different fonts for the Q's in e.g. (4.1) and (8.1)?

Typos: * in the first paragraph on page 2 there is a doubling of "in the" * last paragraph on page 2: deformation(s) * last paragraph on page 2: We then go (to) three * eq. (2.2): fullstop -> comma * eq. (2.4): d -> d/du * above eq. (3.3): This differ(s) * below (3.7): It -> it * below (6.6): is spaces -> in spaces * around (7.6): \check L vs \check\cal L * below (7.8): that (the) this * last sentence of sec. 8: can be compute(d)