Robustness of persistent currents in two-dimensional Dirac systems with disorder

We consider two-dimensional (2D) Dirac quantum ring systems formed by the infinite mass constraint. When an Aharonov-Bohm magnetic flux is present, e.g., through the center of the ring domain, persistent currents, i.e., permanent currents without dissipation, can arise. In real materials, impurities and defects are inevitable, raising the issue of robustness of the persistent currents. Using localized random potential to simulate the disorder, we investigate how the ensemble-averaged current magnitude varies with the disorder density. For comparison, we study the nonrelativistic quantum counterpart by analyzing the solutions of the Schr\"odinger equation under the same geometrical and disorder settings. We find that, for the Dirac ring system, as the disorder density is gradually increased, the average current decreases slowly initially and then plateaus at a finite nonzero value, indicating remarkable robustness of the persistent currents. The physical mechanism responsible for the robustness is the emergence of a class of boundary states---whispering gallery modes. In contrast, in the Schr\"odinger ring system, such boundary states cannot form, and the currents diminish rapidly to zero with an increase in the disorder density. We develop a physical theory based on a quasi-one-dimensional approximation to understand the striking contrast in the behaviors of the persistent currents in the Dirac and Schr\"odinger rings. Our 2D Dirac ring systems with disorder can be experimentally realized, e.g., on the surface of a topological insulator with natural or deliberately added impurities from the fabrication process.