Reflectionless edge states in Dirac models of topological insulators

We study an effective model of topological insulators joined along a bent interface. The Hamiltonian is a Dirac operator $\mathcal{D}=D_1{\sigma }_1+D_2{\sigma }_2+\kappa (x){\sigma }_3$ on $L^2({\mathbb{R}}^2,{\mathbb{C}}^2)$, where $\kappa (x)$ represents a domain wall with a corner: while initially straight, it undergoes a bend when passing from one side of ${\mathbb{R}}^2$ to the other. For most energies in a spectral window centered at $0$, we construct a reflectionless edge state of $\mathcal{D}$: a distorted plane wave with transmission coefficient of modulus $1$. This implies that $\mathcal{D}$ admits modes confined near ${\kappa }^{-1}(\{0\})$, that travel through the bend with no loss neither to the bulk nor to reflection. Our result explains experimental observations from the physics literature, as well as our own numerical simulations.