An integrable discretization of the rational su(2) Gaudin model and related systems

The Gaudin models describe completely integrable classical and quantum long-range interacting spin chains. Originally the Gaudin model was formulated [14] as a spin model related to the Lie algebra sl (2). Later it was realized [15,20] that one can associate such a model with any semi-simple complex Lie algebra g and a solution of the corresponding classical Yang-Baxter equation [5,37]. Depending on the anisotropy of interaction, one distinguishes between XXX, XXZ and XYZ models. Corresponding Lax matrices turn out to depend on the spectral parameter through rational, trigonometric and elliptic functions, respectively. Both the classical and the quantum Gaudin models can be formulated within the r-matrix approach [34]: they admit a linear r-matrix structure, and can be seen as limiting cases of the integrable Heisenberg magnets [39], which admit a quadratic r-matrix structure.

In the 80-es, the quantum rational Gaudin model was studied by Sklyanin [38] and Jurčo [20] from the point of view of the quantum inverse scattering method. Precisely, Sklyanin studied the su(2) rational Gaudin models, diagonalizing the commuting Hamiltonians by means of separation of variables and underlining the connection between his procedure and the functional Bethe Ansatz. In [12] the separation of variables in the rational Gaudin model was interpreted as a geometric Langlands correspondence. On the other hand, the algebraic structure encoded in the linear r-matrix algebra allowed Jurčo to use the algebraic Bethe Ansatz to simultaneously diagonalize the set of commuting Hamiltonians in all cases when g is a generic classical Lie algebra. We have here to mention also the the work of Reyman and Semenov-Tian-Shansky [34]. Classical Hamiltonian systems associated with Lax matrices of the Gaudin-type were widely studied by them in the context of a general group-theoretic approach.

Some others relevants paper on the separability property of Gaudin models are [1,9,10,17,21,39]. In particular, the results in [9,12] are based on the interpretation of elliptic Gaudin models as conformal field theoretical models (Wess-Zumino-Witten models). As a matter of fact, elliptic Gaudin models played an important role in establishing the integrability of the Seiberg-Witten theory [36] and in the study of isomonodromic problems and Knizhnik-Zamolodchikov systems [11,30,35]. Important recent work on (classical and quantum) Gaudin models includes:

• In [10] the bi-Hamiltonian formulation of sl(n) rational Gaudin models has been discussed. A pencil of Poisson brackets has been obtained that recursively defines a complete set of integrals of motion, alternative to the one associated with the standard Lax representation. The constructed integrals coincide, in the sl(2) case, with the Hamiltonians of the bending flows in the moduli space of polygons in the euclidean space introduced in [22]. • In [18] an integrable time-discretization of su(2) rational Gaudin models has been proposed, based on the approach to Bäcklund transformations for finitedimensional integrable systems developed by Sklyanin and Kuznetsov [24]. • Integrable q-deformations of Gaudin models have been considered in [4] within the framework of coalgebras. Also the superalgebra extensions of the Gaudin systems have been worked out, see for instance [7,13,29]. • The quantum eigenvalue problem for the gl(n) rational Gaudin model has been studied and a construction for the higher Hamiltonians has been proposed in [41].

• Recently a certain interest in Gaudin models arose in the theory of condensed matter physics. In fact, it has been noticed [2,33] that the BCS model, describing the superconductivity in metals, and the sl(2) Gaudin models are closely related.

Finally, we mention the so-called algebraic extensions of Gaudin models, which has been studied in [26,27,31] with the help of a general and systematic reduction procedure based on Inönü-Wigner contractions. These extensions constitute also the subject of the present paper, with a slightly different derivation. Suitable algebraic and pole coalescence procedures performed on the Gaudin Lax matrices with N simple poles, provide various families of integrable models whose Lax matrices have higher order poles but share the linear r-matrix structure with the ancestor models. This technique can be applied for any simple Lie algebra g and whatever the dependence (rational, trigonometric, elliptic) on the spectral parameter be. The models characterized by a single pole of increasing order N and with g = su(2), will be called here the one-body su(2) tower. The base of the rational tower (corresponding to N = 2) is nothing but the Lagrange top, a famous integrable system of classical mechanics. The many-body counterpart of

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