Quantum Hall effect in three-dimensional graphene

We theoretically study the electronic band structure and the Hall effect in the negatively-curved three-dimensional (3D) graphene network in magnetic fields. We found that special energy regions appear above and below the zero-energy Landau level, where the 3D Hall conductivity is quantized even though the spectrum is not fully gapped. The energy regions are dominated by the chiral snake states traveling along the zero magnetic field contour on the 3D surface, and the quantization of the Hall conductivity crucially depends on the topology of the contour. The exclusive energy region for the chiral states is found to be a universal feature of 3D graphene systems.