An integrable deformed Landau-Lifshitz model with particle production?

The present article investigates the continuum coherent–state limit of the non-Hermitian integrable deformation of the Heisenberg XXX spin chain that appeared in the classification of $ 4 \times 4 $ Yang–Baxter solutions, referred to as Class 5 and 6. One of the key results is that the Class 5 $R$-matrix is not a genuinely new integrable object, but can be given by a Jordanian Drinfeld twist of the XXX $R$-matrix. In the continuum limit with appropriate scaling of the deformation parameter, the authors derive a leading classical field-theoretic action – non-unitary deformation of the Landau–Lifshitz model (LL). In spin-density basis this yields a Hamiltonian containing a non-diagonalisable “mass/gauge” structure that cannot be rotated to standard anisotropic LL model. Although an explicit Lax pair was not provided, the classical integrability is demonstrated by producing an infinite hierarchy of local conserved charges in involution, generated recursively by continuum analogue of the boost operator. The integrability criterion is based on the boost action as a strong symmetry/recursion operator. Authors point out the emerging symmetries and provide an argument for the on-shell symmetry breaking, i.e. anisotropic dilatation is absent due to the deformation.

By analysing LL scattering in the complex field formulation they find deformation introduces cubic vertices that were absent in the undeformed LL model. A tree-level computation then revealed a nonzero $1 \rightarrow 2$ amplitude, implying a particle decay into two (with soft excitation), which appears to be consistent with FK asymptotic dressing and its relation to Drinfeld twist. Hence authors have established a new example of an integrable deformed model that allows for particle “production”, which is tied to the spin chain non-diagonalisability and $\mathfrak{su}(2)$ breaking down to a single generator, yet preserving integrability. It is indeed an important step towards better understanding of the non-Hermitean integrable models, since it is usually neglected due to its unclear physical properties. In fact, this sector of integrable models and structures emerging therein arise in a number of subfields. For example, it would be particularly important to investigate relation to higher rank LL models, their deformations and lattice realisations, which also admit limits to various dynamical integrable systems (e.g. integrable subsector of Landau-Lifshitz-Gilbert models). Moreover such lower dimensional non-Hermitean structures play a key role in identifying new Poisson algebraic structures on unipotent groups that characterise Hamilton flows in higher quantum simplices. As it was also noted, another crucial question would be to identify a quantisation scheme for strong symmetries and explore how recursive structure translates to the lattice level. The last would be also useful for proving saturation conditions for involutive hierarchies from the quantum minimal surface perspective.

The article is self-consistent and presents clear results, backed by explicit propositions and derivations. I recommend it for publication. Some minor insignificant errata:

1. “many of the unconventional characteristic”, p. 3, paragraph 1
2. “furthers simplifications”, p. 4, paragraph below (1.1)
3. “Galiean boost”, p. 11, footnote 4
4. “once can check”, p. 12, footnote 5
5. “commute among each others”, p. 12, below (2.32)
6. “[R', R]w”, in the last equality of (2.56)
7. “integrable model that do not destroy its integrability”, p. 19, paragraph 3
8. “excitations leaving in the same Jordan chain”, p. 19, paragraph 3
9. “descendants states”, p. 19, paragraph 4
10. “groups others than SU(2)”, p. 20, footnote 14
11. “some constrains”, p. 21, paragraph 1

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We do not believe that the manuscript, in its current state, opens a new pathway in an existing or a new research direction with clear potential for substantial follow-up work. Unless the authors satisfactorily address the critiques raised in the attached referee report, we cannot recommend the publication of the manuscript.

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I recommend that this paper be accepted for publication in this journal.

The authors have written a very interesting paper on an integrable, but non-unitary deformation of the XXX model. They then consider the continuum limit of this theory to find its classical analog. While not quite proving that the classical theory is integrable, they are able to construct an infinite tower of conserved quantities in involution, highly suggesting that the theory is integrable and a Lax pair can be constructed. Perhaps most interesting, they show that the theory can have 1 --> 2 particle production, where one of the two outgoing particles has zero momentum, which is a consequence of momentum conservation.

The paper is clearly written, and as I best as I can tell, free of any errors. I expect there to be a wide interest in the paper. For these reasons I recommend that it be published.

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