Phase singularities and quantum dynamics

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Abstract
We consider Hamiltonians defined on multiply connected domains, which provide models for a great number of physical systems. Their eigenfunctions behave in a characteristic way as a function of enclosed magnetic flux. We establish different possible laws for the change of the phase winding numbers around enclosed flux. These laws provide a common theoretical basis for various quantum effects in orbital magnetism and charged-particle transport. As an illustration we discuss their relevance to particle transport in a constant electric field, emphasizing one-dimensional aspects. We show how the laws of winding-number change directly lead to the equations of quasiclassical dynamics, if the particle is subject to a highly periodic potential, but to completely nonclassical motion if the spatial symmetry is low and the electric field below a threshold value. The nonclassical aspects of quantum transport, particularly Bloch- and Josephson-type oscillations, are due to a periodic appearance of singularities of the phase of the wave function. These singularities are situated at the cores of phase gradient vortices, which periodically move across the physical domain, thereby causing jumps of the phase winding number of each state, associated with a change of the momentum. This picture gives new insight into the microscopic mechanism of momentum change in elastic processes. Further, the speed of motion of the phase singularities determines whether a state is insulating or conducting. We thus obtain a new characterization of such states.