Surface electron currents are studied for the integer quantum Hall effect in three dimensions (3D) proposed previously by Kohmoto et al. [Phys. Rev. B $45,$ 13 488 (1992)] and by Koshino et al. [Phys. Rev. Lett. 86, 1062 (2001)]. We predict that the current wraps the facets of the sample with its intensity and direction dictated by 3D Chern numbers that are just the quantized Hall conductivities in 3D $({\sigma }_{\mathrm{xy}},{\sigma }_{\mathrm{zx}}),$ so a natural connection exists between the surface and bulk currents just as in 2D. An experiment to detect the 3D integer quantum Hall effect through the wrapping current is proposed.

• Received 8 April 2002

DOI:https://doi.org/10.1103/PhysRevB.66.081301