An edge state is a time-harmonic mode of a conservative wave system, e.g. Schroedinger, Maxwell, which is propagating (plane-wave-like) parallel to and localized, transverse to a line-defect or “edge”. Topologically protected edge states are edge states which are immune to local scattering impurities.
First studied in the context of the quantum Hall effect, protected edge states have attracted great interest recently due to their role in the field of topological insulators. Such states are potential vehicles for robust energy-transfer in the presence of strong localized defects and random imperfections. They are therefore considered ideal for use in nano-scale devices.
The theoretical understanding of topological protection has mainly come from discrete (tight-binding) models and direct numerical simulation. After an introduction to the spectral properties of continuous honeycomb structures and their novel properties such as Dirac points, we introduce a rich family of continuum PDE models and discuss regimes (“phases") where topologically protected edge states exist along a “zig-zag edge", and regimes where edge states may exist but are not protected. These results follow from a general theorem on the bifurcation of edge states from Dirac points of the background honeycomb structure. This bifurcation is seeded by the zero mode of an effective Dirac equation. The key to applying the general theorem is the verification of a spectral no-fold condition along the zig-zag edge.
This is joint work with C.L. Fefferman and J.P. Lee-Thorp.