Quantum Phase Transition in Finite-Size Lipkin-Meshkov-Glick Model
  Lipkin model of arbitrary particle-number N is studied in terms of exact differential-operator representation of spin-operators from which we obtain the low-lying energy spectrum with the instanton method of quantum tunneling. Our new observation is that the well known quantum phase transition can also occur in the finite-N model only if N is an odd-number. We furthermore demonstrate a new type of quantum phase transition characterized by level-crossing which is induced by the geometric phase interference and is marvelously periodic with respect to the coupling parameter. Finally the conventional quantum phase transition is understood intuitively from the tunneling formulation in the thermodynamic limit. 
Gang Chen, J.-Q.Liang