Classical and quantum phase transitions in the Lipkin-Meshkov-Glick model

An analysis of the classical and quantum phase transitions of the Lipkin-Meshkov-Glick model is presented. It is shown that the classical dynamics is ruled by the energy surface of the system. Applying the catastrophe formalism to this energy surface the separatrix is obtained. It determines the regions in the control parameter space where there are phase transitions. Special attention is given to the compositions of ground and first-excited energy states, which are well described by the even and odd SU(2) coherent states. Phase transitions are shown to be associated with a change in the wave functions from collective to single-particle behavior. Evaluating the distribution of nearest-neighbor spacings it is shown that the separatrix of the system emerges as a useful tool to describe the global behavior of the quantum-level structure and their corresponding wave functions