Supersymmetric structure of quantum Hall effects in graphene

Electrons behave as Dirac fermions in graphene, though their speed is given by the Fermi velocity. In such a system the Zeeman splitting is exactly as large as the Landau level separation. It leads to the emergence of the zero-energy state and multiplets made of the nonzero-energy up-spin and down-spin states. Hence, the supersymmetry is a good symmetry in graphene. We present a unified description of quantum Hall effects in multilayer graphene based on the supersymmetric formalism. We extend the Dirac Hamiltonian to include two indices j↑j↑ and j↓j↓, characterized by the dispersion relation E(p)∝pj↑+j↓E(p)∝pj↑+j↓ and the Berry phase π(j↑−j↓)π(j↑−j↓). The quantized Hall conductivity is shown to be σxy=±(2N+j↑+j↓)2e2/hσxy=±(2N+j↑+j↓)2e2/h for N=0,1,2,3,….