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

Abstract

We show that the Lipkin-Meshkov-Glick 2𝑁-fermion model is a particular case of a one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of a 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 an 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.