  Using the Wherl entropy, we study the delocalization in phase-space of energy
eigenstates in the vicinity of avoided crossing in the Lipkin-Meshkov-Glick
model. These avoided crossing, appearing at intermediate energies in a certain
parameter region of the model, originate classically from pairs of trajectories
lying in different phase space regions, which contrary to the low energy
regime, are not connected by the discrete parity symmetry of the model. As
coupling parameters are varied, a sudden increase of the Wherl entropy is
observed for eigenstates close to the critical energy of the excited-state
quantum phase transition (ESQPT). This allows to detect when an avoided
crossing is accompanied by a superposition of the pair of classical
trajectories in the Husimi functions of eigenstates. This superposition yields
an enhancement of dynamical tunneling, which is observed by considering initial
Bloch states that evolve partially into the partner region of the paired
classical trajectories, thus breaking the quantum-classical correspondence in
the evolution of observables.
