The Lipkin-Meshkov-Glick model as a particular limit of the SU(1,1) Richardson-Gaudin integrable models

TL;DR

This paper demonstrates that the Lipkin-Meshkov-Glick model is a specific case of SU(1,1) Richardson-Gaudin integrable models, using exact solutions to analyze spectral parameters and phase transitions.

Contribution

It reveals the LMG model as a special limit of SU(1,1) RG models and develops a numerical method to study pairons across different phases and transitions.

Findings

Spectral parameters (pairons) determine wave functions in various phases.

Insights into first, second, and third order phase transitions.

Analysis of spectral crossings in the LMG spectrum.

Abstract

The Lipkin-Meshkov-Glick (LMG) model has a Schwinger boson realization in terms of a two-level boson pairing Hamiltonian. Through this realization, it has been shown that the LMG model is a particular case of the SU (1, 1) Richardson-Gaudin (RG) integrable models. We exploit the exact solvability of the model tostudy the behavior of the spectral parameters (pairons) that completely determine the wave function in the different phases, and across the phase transitions. Based on the relation between the Richardson equations and the Lam\'e differential equations we develop a method to obtain numerically the pairons. The dynamics of pairons in the ground and excited states provides new insights into the first, second and third order phase transitions, as well as into the crossings taking place in the LMG spectrum.