Engineering Supersymmetric Potentials on the Edge of a Quantum Spin Hall System: Emergence of Volkov-Pankratov States and Construction of Reflectionless Potentials

Volkov-Pankratov (VP) states are a family of sub-gap states which appear at the smooth interface ( domain wall ) which separates two distinct gapped topological states of matter. We exploit known results from supersymmetry quantum mechanics to study the emergence of such states in the edge spectrum of the quantum spin Hall edge state when exposed to a smoothly varying mass term (Zeeman field) that switches sign at a given spatial point, thereby inducing band-inversion. Both the VP states at non-zero energy and the zero energy Jackiw-Rebbi mode stay localized at the interface, however the spin texture of the VP states turns out to be endowed with periodic windings in real space which is controllable with an electric field to which the Jackiw-Rebbi mode is insensitive. Moreover, the VP states exhibit an intriguing interplay between the electric and the magnetic field with a collapse of the entire spectrum when they are equal. Our theoretical predictions are confirmed with a BHZ model lattice simulation¹. We further take a tan hyperbolic type mass term (which is experimentally more realizable) and show the existence of the in-gap bound states. This type of potential also has the interesting property of reflectionless transmission when certain conditions are met. Using this idea, one can engineer such potentials in the BHZ lattice model in a quantum transport simulation to study the above the gap transmission which is very close to unity. This may have a potential application to engineer a device with lossless transmission².