Investigation of topological edge states using generalized Bloch theorem - Abhijeet Alase

The non-trivial topology of the bulk of topological insulators and superconductors manifests in the form of edge/surface states protected by discrete symmetries - a principle known as the "bulk-boundary correspondence". While our understanding of such protected states has so far largely relied on the analysis of the bulk topological properties, a main drawback of this approach is that it provides little (if any) information about the wavefunctions of the protected states, as well as their response to symmetry-breaking perturbations. We present a generalization of Bloch's theorem able to accommodate the effects of terminations and interfaces modeled by effective boundary conditions, leading to an exact algebraic solution for all energy eigenstates and eigenvalues. Using our approach, we compute analytically the energy eigenvalues and eigenstates of Su-Schrieffer- Heeger model and Kitaev's Majorana chain. We show that for certain parameter values, the Majorana modes decay exponentially in space with a power-law prefactor. We also show that the penetration depth of the chiral edge states of p+ip superconductor on a lattice diverges near those points in the surface Brillouin zone where the surface bands touch the bulk bands.