Delocalization properties at isolated avoided crossings in Lipkin-Meshkov-Glick type Hamiltonian models

We present a detailed study of the behavior of the delocalization properties in the phase space of the Husimi function of ground and excited states in the avoided crossings vicinity for a Hamiltonian of Lipkin-MeshkovGlick type. The analysis was done numerically with a calculation of the second moment of the Husimi function and the Wehrl entropy for the Hamiltonian eigenstates. Avoided crossings have been determined by calculating the real part of the exceptional points imposing a threshold value for the energy difference between adjacent levels. We have found that the behavior of these quantities indicates locatization-delocalization in phase space for adjacent energy levels in avoided crossings. The obtained results have been further explained by a perturbative calculation of the derivative of the second moment of the Husimi function with respect to the control parameter.