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

TL;DR

This paper presents an exactly solvable analytical model for the interface between trivial and topological 2D insulators, revealing how the number of bound states and reflection properties depend on measurable parameters.

Contribution

It introduces a phenomenological model based on the Schrödinger equation with a modified Pöschl-Teller potential, linking physical parameters to bound states and reflection behavior.

Findings

Number of bound states determined by integer part of a dimensionless parameter

Wave reflection is absent when the parameter is an integer

Model parameters are relevant to typical condensed matter systems

Abstract

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.