Four-band insulator on a $\mathbb{Z}_2$ domain wall: an analytically solvable model for the interface between trivial and topological 2D insulators

A phenomenological model for the interface between trivial and topological two-dimensional insulators possessing the same band gap is presented. The model depends on three measurable parameters, the energy gap $E_g$, the Fermi velocity of the metallic edge states $v_F$ and the thickness of the interface $\Delta$ where the gap inversion occurs, and can be reduced to the Schr\"odinger equation for the modified P\"oschl-Teller potential, which admits an analytical solution. It is demonstrated that the underlying physics is determined by the adimensional parameter $\alpha=E_g\Delta/2\hbar v_F$, whose integral part determines the number of massive bound states at the interface. Furthermore, when $\alpha$ is exactly an integer, waves incident on the interface are never reflected. Results for parameters chosen in the typical scale of condensed matter systems are briefly discussed.