Lifting integrable models and long-range interactions

1 - The paper addresses an interesting long standing question of long range spin chains.
2 - The paper is well written.

1 - Some definitions are mathematically not rigorous.

The paper addresses a long-standing problem, namely whether there are R-matrices and Lax-operators for the perturbative long-range spin chains which appear in the AdS/CFT correspondence. A compact definition for the perturbative long-range spin chains on the level of charges has been available since 2008. This definition made it possible to determine the asymptotic spectrum, but the disadvantage is that the charges are defined only for infinite length. The Lax operators of spin chains have proven to be very useful for nearest-neighbor interactions, but their generalization for long-range deformations has not been determined yet. This article tries to fill this gap.

The authors generalized the definitions of [14] for perturbative long range deformations. It gives the perturbative definitions of the Lax-operators and R-matrices for the long range spin chains. The long range deformations of the six vertex models have already been classified on the level of charges and the authors demonstrated that there exist Lax-operators and R-matrices for all integrable deformations of the six vertex models in the first order. They also determined the Lax-operator of the dilatation operators of the SU(2) sector in N=4 SYM in three loops.

The article contains significant advances in an important and persistent problem of the AdS/CFT duality, so I recommend it for publication after the below points have been addressed.

See the attached Requested_changes.pdf

1. innovative
2. rigorous

n/a

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.

The paper at hand discusses the interesting conceptual question in how far integrability formulated via conserved charges can be lifted to Lax operators or R-matrices. This paper should be considered in the context of the letter referred to as [16], which appeared at the same time and discusses similar aspects in a systematic way. The discussed situations and examples are useful to demonstrate that indeed in many cases on can find Lax operators and R-matrices that obey Yang-Baxter relations and correspond to the considered charge operators. The implications of the presented findings are not completely conclusive and the usefulness of the discussed results remains a bit opaque which is reflected in the very short section 8. E.g. it is stated that the results "should open the door for applications of the algebraic Bethe Ansatz" but this is not demonstrated explicitly. In particular, the question of wrapping is not addressed as opposed to reference [16] which is one of the most interesting points in the context of long-range spin chains. Still the result provides useful insight into the algebraic structure of integrable systems and progresses a question that has been on the table since the discovery of AdS/CFT type long-range spin chains. I therefore recommend it for publication after the below points have been addressed.

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)