We exploit an index theorem and a high degree of symmetry to understand unusual quantum Hall effects of the n = 0 Landau level in graphene. The high symmetry such as the pseudospin symmetry of Fermi points in a graphene sheet cannot couple to an external magnetic field. In the absence of the magnetic field, the index theorem provides a relation between the zero-energy state of the graphene sheet and the topological deformation of the compact lattice. Under the topological deformation, the zero-energy state emerges naturally without the Zeeman splitting at the Fermi points in the graphene sheet. This results in the fact that the pseudospin is an exact symmetry. In the case of nonzero energy, the up-spin and down-spin states have the exact high symmetry of spin, forming the pseudospin singlet pairing. We describe the peculiar and unconventional quantum Hall effects of the n = 0 Landau level in graphene on the basis of the index theorem and the high degree of symmetry.
Zero-Energy State in Graphene in a High Magnetic Field
