Theoretical and Mathematical Physics, 189(1): 1509–1527 (2016) “TWISTED” RATIONAL r-MATRICES AND THE ALGEBRAIC BETHE ANSATZ: APPLICATIONS TO GENERALIZED GAUDIN MODELS, BOSE–HUBBARD DIMERS, AND JAYNES–CUMMINGS–DICKE-TYPE MODELS T. V. Skrypnyk ∗ We construct quantum integrable systems associated with the Lie algebra gl(n) and non-skew-symmetric “shifted and twisted” rational r-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, n-level (n−1)-mode Jaynes–Cummings–Dicke-type models in the Λ- configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz. Keywords: integrable system, classical r-matrix, algebraic Bethe ansatz DOI: 10.1134/S004057791610010X 1. Introduction Quantum integrable models with long-range interactions play an important role in nonperturbative physics. They are used in the theory of small quantum grains [1]–[4], where the famous Richardson model [5] is used, in nuclear physics [6], [7], in the theory of colored Fermi gases [8], in the theory of Bose–Einstein condensates [9], [10], in the so-called central spin model theory [11], [12], and so on. The main examples of such integrable models are Gaudin models [13], Gaudin models in external mag- netic fields [14], Bose–Hubbard dimer models [9], [10], ordinary “two-level one-mode” Jaynes–Cummings– Dicke models [15], [16] and their n-level many-mode generalizations [17], [18]. The common feature of these models is their connection with skew-symmetric classical r-matrices [14], [19], but the approach to quan- tum integrable systems based on skew-symmetric classical r-matrices has strong limitations. Indeed, by the Belavin–Drinfeld theorem [20], skew-symmetric classical r-matrices can only be rational, trigonometric, or elliptic. Moreover, elliptic r-matrices are not Cartan-invariant, and the corresponding integrable systems have no “particle number conservation operators.” This restricts their physical applications. Furthermore, the so-called “nonstandard” rational [21] and “nonstandard” trigonometric skew-symmetric r-matrices [20] produce non-Hermitian Hamiltonians, which makes it almost impossible to apply them in quantum theory. Therefore, the role of skew-symmetric classical r-matrices (as applied to the theory of quantum integrable systems) mainly reduces to two classical r-matrices: standard rational and standard trigonometric ones. In a series of previous papers [22]–[26], we proposed associating quantum integrable models with non- skew-symmetric classical r-matrices that are outside the Belavin–Drinfeld classification [20]. In particular, we thus constructed the generalization of the Gaudin models [22] corresponding to any simple (reductive) Lie * University of Milano-Bicocca, Milano, Italy; Bogoliubov Institute for Theoretical Physics, Kiev, Ukraine, e-mail: taras.skrypnyk@unimib.it. Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 1, pp. 125–146, October, 2016. 0040-5779/16/1891-1509 c  2016 Pleiades Publishing, Ltd. 1509