Coulomb drag in topological materials

Dirac fermions are at the forefront of modem condensed matter physics research. They are known to occur in materials as diverse as graphene, topological insulators, and transition metal dichalcogenides, while closely related Weyl fermions have been discovered in other materials. They have been predicted to lend themselves to a variety of technological applications, while the recent prediction and discovery of the quantized anomalous Hall effect of massive Dirac fermions is regarded as a potential gateway towards low-energy electronics. Some materials hosting Dirac fermions are natural platforms for interlayer coherence effects such as Coulomb drag and exciton condensation. The top and bottom surfaces of a thin topological insulator film provide such a prototype system. Here we describe recent insights into Coulomb drag between two layers of Dirac fermions relying primarily on topological insulator films as a minimal model. We consider both non-magnetic topological insulators, hosting massless Dirac fermions, and magnetic topological insulators, in which the fermions are massive. We discuss in general terms the dynamics of the thin-film spin density matrix, outlining numerical results and approximate analytical expressions where appropriate for the drag resistivity rho(D) at low temperatures and low electron densities. In magnetic topological insulators with out-of-plane magnetizations in both the active and passive layers we analyze the role of the anomalous Hall effect in Coulomb drag. Whereas the transverse response of the active layer is dominated by a topological term stemming from the Berry curvature, we show that neither the topological mechanism nor disorder renormalizations associated with it contribute to Coulomb drag. Nevertheless, the longitudinal drag force in the passive layer does give rise to a transverse drag current that is independent of the active-layer magnetization. It depends non-monotonically on the passive-layer magnetization, exhibiting a peak that becomes more pronounced at low densities. All of these observations can be verified in the laboratory. We compare results for topological insulators with results for graphene, identifying qualitative and quantitative differences, and discuss generalisations to multi-valley systems, ultra-thin films and electron-hole layers.