Wrapping Current versus Bulk Integer Quantum Hall Effect in Three Dimensions

Surface electron currents are studied for the integer quantum Hall effect in three dimensions(3D) proposed previously by Kohmoto et al. and by Koshino et al. We predict the current wraps the facets of the sample with its intensity and direction dictated by 3D Chern numbers which are just the quantized Hall conductivities in 3D ($\sigma_{xy}, \sigma_{zx}$), 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.