Exact solutions of the Lipkin-Meshkov-Glick model Hamiltonian based on the Bethe ansatz

Dongyang Yu

Hiroyuki Morita, Hiromasa Ohnishi, João da Providência +1 more

T. Skrypnyk

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.

L.V. Avdeev, Alexey A. Vladimirov

Pierre Nataf

The authors find that the SU($N$) Fermi-Hubbard model on two sites can be mapped to the Lipkin-Meshkov-Glick model. It admits a Bethe ansatz solution and exhibits a quantum phase transition in the large-$N$ limit that could be experimentally accessible using current technologies involving SU($N$) ultracold atoms or molecules.

Z. ALIPOUR, M. A. Jafarizadeh, N. FOOLADI +1 more

Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.