The deformed Inozemtsev spin chain

Hereby we submit our revised manuscript. All feedback of the referees, for which we are very grateful, has been taken into account.

There is one point raised by all three referees: that we do not give the proof of integrability of our elliptic dynamical spin-Ruijsenaars (in the sense that our hierarchy of matrix-valued difference operators commute), and therefore of our deformed Inozemtsev chain. As we mention, our proof of this fact is highly technical; much more so than the corresponding proof of Matushko and Zotov in Ref. [45], which unfortunately does not seem to extend to our case. Our proof warrants a separate publication, which will be much more mathematical. We firmly believe that in addition to that proof it is important to present the concrete results to a more physics-inclined audience. This is the aim of the present paper.

We have changed the notation for the isotropic hamiltonians to \bar{H}, in accordance with our use of \bar{V} for the isotropic potentials. Hence:
p2, preceding Eq. (1): "with hamiltonian of the form" -> "Using a bar to denote isotropic case, these spin chains have hamiltonian of the form"

p3, Fig. 1: written out "anisotr." -> "anisotropic"
p3, first paragraph: "conjecture for the conserved charges" -> "conjecture for a hierarchy of conserved charges" to clarify what we loosely speaking mean by 'integrable'
Just below: Footnote 1 added, with further clarification of this point, and its connection to the recent preprint of Chalykh (new Ref [24]).

p4, Outline: moved the sentence + footnote "While we focus on spin 1/2, [...]" to the bottom of outline.
p4, preceding Eq. (5): corrected "periodisation" -> "periodic version"
p4, following Eq. (6): clarified "see (A.6)" -> "and in '~' we omit some constants, see (A.6) for the precise relation"

p6, top, following Eq. (7): added "where the `^\star' serves to distinguish these fixed parameters from their unrestricted counterparts x_k that will appear in \textsection 3"
p6, Sect 2.2: clarified "The spectrum is real" -> "While the hamiltonians are not hermitian for the standard scalar product, numerics shows that the spectrum is real"
p6, Defining properties, isotropic limit: added reference "(see \textsection C)" for details of the computation
p6, Defining properties, long-range limit:
added "To see this use \theta(x) \to N \sin(\pi x/N)/\pi as \kappa \to 0."
rephrased "The potential (6) has long-range limit" -> "The potential (6) thus has long-range limit"
p7, top: rephrased "If moreover \eta a \to -i \infty" -> "When \kappa\to 0 and moreover \eta a\to - i \infty$ for fixed \eta"

p7, following Eq. (19): added explanation "Namely, G^N = K_1 ... K_N$ is a central element, [...] These are the simplest eigenstates after the reference vector \ket{\uparrow ... \uparrow}."
Just below: rephrased "We have not yet been able to find an expression for the dispersion relation" -> "We have not yet been able to find
a compact expression for their chiral dispersion relations."

p8, Eq. (21): added missing parenthesis

p9, Choice of R-matrix: corrected \eqref from 2.2 to (10)
Just below: added Ref [42] to Baxter

p11, Modular family: corrected Ref [40]

p17, following Eq. (A.4): added "When \kappa\to 0 we have \theta(x) \to N \sin(\pi x/N)/\pi by the Jacobi imaginary transformation."

p20, end of App C: added details for isotropic limit "To evaluate the isotropic limit of E(x,a) [...]."

References: corrected capitalisation, added arXiv and doi data