Persistent currents and edge states in a magnetic field

We investigate equilibrium electron currents in an ideal two-dimensional ring (of radii ${\mathit{R}}_1$${\mathit{R}}_2$). The most striking result emerges when the conditions for the existence of edge and bulk states are met, namely ${\mathit{R}}_2$-${\mathit{R}}_1$≫${\mathit{a}}_{\mathit{H}}$ where ${\mathit{a}}_{\mathit{H}}$ is the magnetic length. If the Fermi energy lies in a gap between two Landau levels, the current (as a function of electron density) displays violent fluctuations (in sign and in absolute value), which is quite unusual for systems without disorder. The fluctuations in sign result from the alternative contributions of inner and outer occupied edge states below the Fermi energy, while those in absolute value originate from the apparent asymmetry between the slopes of the energy curves near the two opposite edge states. On the other hand, when the Fermi energy is locked on a Landau level, the current has a plateau as a function of electron density. Its value at a plateau represents the contribution to the current of all the edge states in the lower Landau levels.