Topological insulators in semiclassical regime

Abstract

We study solutions of $2 \times 2$ systems $(h D_t + \mathcal{D}) \Psi_t = 0$ on $\mathbb{R}^2$ in the semiclassical regime $h \rightarrow 0$. Our Dirac operator $\mathcal{D}$ is a standard model for interfaces between topological insulators: it represents a semimetal at null energy, and distinct topological phases at positive / negative energies. Its semiclassical symbol has two opposite eigenvalues, intersecting conically along the curve $\Gamma$ of positions / momenta where the material behaves like a semimetal. We prove that wavepackets solving $(h D_t + \mathcal{D}) \Psi_t = 0$, initially concentrated in phase space on $\Gamma$, split in two parts. The first part travels coherently along $\Gamma$ with predetermined direction and speed. It is the dynamical manifestation of the famous "edge state". The second part immediately collapses. Our approach consists of: (i) a Fourier integral operator reduction; (ii) a careful WKB analysis of a canonical model; (iii) a reconstruction procedure. It yields concrete formulas for the speed and profile of the traveling modes. As applications, we analytically describe dynamical edge states for models of magnetic, curved, and strained topological insulators.