Diffusion in the random gap model of mono- and bilayer graphene

In this paper we study the effect of a fluctuating gap in mono- and bilayer graphene, created by a random staggered potential. We identify a continuous symmetry for the two-particle Green's function which is spontaneously broken in the average two-particle Green's function and leads to a massless fermion mode. Within a loop expansion it is shown that the massless mode is dominated on large scales by small loops. This result indicates diffusion of electrons. Although the diffusion mechanism is the same in mono- and in bilayer graphene, the amount of scattering is much stronger in the latter. Physical quantities at the neutrality point, such as the density of states, the diffusion coefficient and the conductivity, are determined by the one-particle scattering rate. All these quantities vanish at a critical value of the average staggered potential, signaling a continuous transition to an insulating behavior.