Topological Delocalization of Two-Dimensional Massless Dirac Fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which, e.g., underlies theoretical descriptions of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing conductance. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of the states of the massless Dirac Hamiltonian can be localized.

Figure 1

The beta functions of the random Dirac Hamiltonian (2) (closed circles) and the random spin-orbit model (6) (open circles). The broken line represents the one-loop beta function of the conventional 2D symplectic class, $\beta (g)\sim 1/\pi g$ [12].

Figure 2

The evolution of energy spectra as a function of the twist angle $\varphi$ for the random massless Dirac model (left) and the random spin-orbit model (right).