Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 074, 18 pages      arXiv:2210.04454      https://doi.org/10.3842/SIGMA.2022.074

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

Taras Skrypnyk
Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv, 03680, Ukraine

Received June 19, 2022, in final form September 16, 2022; Published online October 10, 2022

Abstract
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 $N=1,2$.

Key words: classical $r$-matrix; Gaudin-type model; algebraic Bethe ansatz.

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