Exact edge, bulk, and bound states of finite topological systems

Abstract

Finite topologically non-trivial systems are often characterised by the presence of bound states at their physical edges. These topological edge modes can be distinguished from usual Shockley waves energetically, as their energies remain finite and in-gap. On a clean 1D or reducible 2D model, in either the commensurate or semi-infinite case, the edge modes can be obtained analytically, as shown in [PRL 71, 3697 (1993)] and [PRA 89, 023619 (2014)]. We put forward a method for obtaining the spectrum and wave functions of topological edge modes for arbitrary finite lattices, including the incommensurate case. A small number of parameters are easily determined numerically, with the form of the eigenstates remaining fully analytical. We also obtain the bulk modes in the finite system analytically and their eigenenergies, which lie within the infinite-size limit continuum. Our method is general and can be easily applied to obtain the properties of non-topological models and/or extended to include impurities. As an example, we consider the case of an impurity located next to one edge of a 1D system, equivalent to a softened boundary in a separable 2D model. We show that a localised impurity can have a drastic effect on the edge modes of the system. Using the periodic Harper and Hofstadter models to illustrate our method, we find that, on increasing the impurity strength, edge states can enter or exit the continuum, and a trivial Shockley state bound to the impurity may appear. The fate of the topological edge modes in the presence of impurities can be addressed by quenching the impurity strength. We find that at certain critical impurity strengths, the transition probability for a particle initially prepared in an edge mode to decay into the bulk exhibits discontinuities that mark the entry and exit points of edge modes from and into the bulk spectrum.