The Generalized Lipkin–Meshkov–Glick Model and the Modified Algebraic Bethe Ansatz

We show that the Lipkin–Meshkov–Glick $2N$-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic $r$-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin–Meshkov–Glick fermion model based on the Gaudin-type model corresponding to the same $r$-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky–Volterra gyrostat. We diagonalize the corresponding quantum Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly solve the corresponding Bethe-type equations for the case of small fermion number $N=1,2$.