Multifractal zero mode for disordered graphene

Off-diagonal disorder with random hopping between the sublattices of a bipartite lattice is described by a Hamiltonian which has chiral (sub-lattice) symmetry. The energy spectrum is symmetric around E=0 and for odd total number of lattice sites an isolated zero mode always exists, which coincides with the mobility edge of an Anderson transition in two dimensions(2D). In the chiral orthogonal symmetry class BDI we compute the fractal dimension $D_{2}$ of the zero mode for graphene samples with edges. In the absence of disorder $D_{2}=1$, which corresponds to a one-dimensional edge states, while for strong disorder $D_{2}$ decays towards 0 and the zero mode becomes localized. The similarities and differences between zero modes in the honeycomb and the square bipartite lattices are pointed out.

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