Exact edge, bulk and bound states of finite topological systems

TL;DR

This paper presents a general analytical method to determine the spectrum and wave functions of topological edge and bulk states in finite systems, including incommensurate and impurity-affected cases, with applications to models like Harper and Hofstadter.

Contribution

A novel analytical approach for finite topological systems that accurately characterizes edge and bulk states, including effects of impurities and incommensurate structures.

Findings

Edge states can be analytically obtained in finite systems.

Impurities can significantly alter edge state properties.

Critical impurity strengths cause topological edge modes to merge with bulk spectrum.

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.