Edge states of a 1D topological insulator

Abstract

One of the most famous quasi particles is the exciton, a bound state of an electron and an electron hole. It is still being intensively researched, because it is very interesting. The exciton is an excitation of a topological insulator. We want to know all the properties of the edge states of topological insulators in one dimension. More specifically, we want to know the wave function and its energy. We found that edge states always have the same energy as the energy halfway the band gap. And that the wave functions peak at the topological side of the edge. We also found that we can rewrite the Hamiltonian to that of the Zeeman interaction, from which we can deduce the topological class of the Hamiltonian. The difference in topological classes explains why the edge states always have the same energy as halfway the band gap. Because the boundary between the topological classes has that energy. By researching the properties of edge states of a topological insulators in one dimension, we can obtain more knowledge about excitons and their edge states in the future.