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

Finite topologically nontrivial systems are characterized, among many other unique properties, 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 even when the boundaries of the system represent an effectively infinite and sharp energetic barrier. Theoretically, the existence of topological edge modes can be shown by means of the bulk-edge correspondence and topological invariants. On a clean one-dimensional lattice and reducible two-dimensional models, in either the commensurate or semi-infinite case, the edge modes can be essentially obtained analytically, as shown previously [Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993); D. Hugel and B. Paredes, Phys. Rev. A 89, 023619 (2014)]. In this work, 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 associated eigenenergies, which lie within the infinite-size limit continuum. Our method is general and can be easily applied to obtain the properties of nontopological models and/or extended to include impurities. As an example, we consider a relevant case of an impurity located next to one edge of a one-dimensional system, equivalent to a softened boundary in a separable two-dimensional model. We show that a localized impurity can have a drastic effect on the original topological 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.