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

摘要

The redistribution of energy levels between energy bands is studied for afamily of simple effective Hamiltonians depending on one control parameter andpossessing axial symmetry and energy-reflection symmetry. Further study is madeon the topological phase transition in the corresponding semi-quantum andcompletely classical models, and finally the joint spectrum of the twocommuting 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 toexist among the redistribution of energy levels between energy bands for thequantum Hamiltonian, the modification of Chern numbers of eigenline bundles forthe corresponding semi-quantum Hamiltonian, and the presence of Hamiltonianmonodromy for the complete classical analog. In particular, as far as the bandrearrangement is concerned, a fine agreement is found between theredistribution of the energy levels described in terms of joint spectrum ofenergy and momentum in the full quantum model and the evolution ofsingularities of the energy-momentum map of the complete classical model. Thetopological phase transition observed in the present semi-quantum and thecomplete classical models are analogous to topological phase transitions ofmatter.