Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models

The redistribution of energy levels between energy bands is studied for a family of simple effective Hamiltonians depending on one control parameter and possessing axial symmetry and energy-reflection symmetry. Further study is made on the topological phase transition in the corresponding semi-quantum and completely classical models, and finally the joint spectrum of the two commuting observables $(H=E,J_z)$ (also called the lattice of quantum states) is superposed on the image of the energy-momentum map for the classical model. Through these comparative analyses, mutual correspondence is demonstrated to exist among the redistribution of energy levels between energy bands for the quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian monodromy for the complete classical analog. In particular, as far as the band rearrangement is concerned, a fine agreement is found between the redistribution of the energy levels described in terms of joint spectrum of energy and momentum in the full quantum model and the evolution of singularities of the energy-momentum map of the complete classical model. The topological phase transition observed in the present semi-quantum and the complete classical models are analogous to topological phase transitions of matter.