The surface of a topological insulator is a closed two-dimensional manifold. The surface states are described by the Dirac Hamiltonian in curved two-dimensional spaces. For a slablike sample with a magnetic field perpendicular to its top and bottom surfaces, there are chiral states delocalized on the four side faces. These “chiral sheets” carry both charge and spin currents. In strong magnetic fields, the quantized charge Hall effect $[{\sigma }_{xy}=(2n+1)e^2/h]$ will coexist with spin Hall effect.

• Received 29 August 2009

DOI:https://doi.org/10.1103/PhysRevLett.103.196804