Study On Topological Properties Of Wave Propagation In One-dimensional Periodic System

The topological properties have long been used to characterize the novel states of the materials. For example the integer quantum Hall effect can be related to the topological properties of the system and the quantum Hall conductance can be related to the Chern number of occupied energy bands of the electronic system. The discovery of topological insulators has fueled a surge of interests on the topological properties of the two and three dimensional systems and become a hot area of physical research. Like normal insulators, topological insulators also have bulk energy gap but gapless states at the surface. Under the influences of magnetism and superconductivity a lot of novel quantum phenomena are predicted, which may find applications in spintronics and quantum computing. Recently it’s proposed that one dimensional system can also be used to study the nontrivial topological phases and gapless edge states. To explore the topological phases of the one dimensional topological properties this dissertation contains two parts of work: theoretical study of effect of disorders on the topological phase of the one dimensional optical superlattice and the experimental study of the gapless edge states in the one dimensional classical string with periodic density.The theoretical background and the basic concepts of the wave propagation is introduced first in the dissertation. Then three numerical methods for calculating the eigenvalues and eigenfunctions of the one dimensional periodic system are discussed: orthogonal function method, finite difference method and tight-binding method. Both the periodic and open boundary conditions are used in the above three methods. The periodic boundary condition is mainly used to study the bulk energy and frequency bands. The open or fixed boundary condition suit for the studying the edge states of the one dimensional system. Based on these three numerical methods we demonstrate how the topological invariants the one dimensional system are calculated numerically.Using these numerical methods we study the effect of the disorders on the gapless edge states and density plateaus of the one dimensional superlattice popularized with cold atoms in detail. As is pointed out in the PRL paper of Lang Lijun,the topological phases of the one dimensional superlattice can exhibit themselves through the existence of nontrivial edge states and topological density plateaus. We study the model in paper PRL which can be mapped to a two dimensional model. The energy bands of this two dimensional system have nonzero Chern numbers. So this two dimensional system topologically equivalent to a quantum Hall state. There are gapless edge at the bulk energy gaps. When disorders are present the bulk bands are broaden and the bulk gaps are shrunk by the random potentials. However, as long as the bulk gaps remains open the edge states are still gapless. So the we confirm that the gapless edge states are topological protected against the disorders. The effects of disorders on the density plateaus defined in PRL are also simulated. The numerical simulations show that the widths of the density plateaus depend on the width of the corresponding bulk energy gaps. The results also show the disorders don’t alter this conclusion. When the strength of the disorders increases the widths of the bulk gaps decreases and the width of the density plateaus decreases with the bulk gaps. The numerical results show that they are approximately proportional to each other. When the disorders are strong enough, the bulk gaps close and the density plateaus disappear. However, with weak disorders the density plateaus are robust when we using a longer interval to define the local density. So we confirm that the density plateaus are topological properties of the one dimensional system.Experimentally we measure the spectrum of the gapless edge states. It’s theoretically found that classical string with periodic density has frequency bands similar to the energy bands of the electrons in the one dimensional periodic system. According to the bulk and edge correspondence there must be gapless edge states in the bulk frequency gaps. Our experimental results are in good agreement of the theoretical simulations, which are the first direct measurement of the spectrum of the topologically protected gapless edge state.