Finite-size scaling analysis of the conductivity of Dirac electrons on a surface of disordered topological insulators

Abstract

Two-dimensional (2D) massless Dirac electrons appear on a surface of three-dimensional topological insulators. The conductivity of such a 2D Dirac electron system is studied for strong topological insulators in the case of the Fermi level being located at the Dirac point. The average conductivity $\langle\sigma\rangle$ is numerically calculated for a system of length $L$ and width $W$ under the periodic or antiperiodic boundary condition in the transverse direction, and its behavior is analyzed by applying a finite-size scaling approach. It is shown that $\langle\sigma\rangle$ is minimized at the clean limit, where it becomes scale-invariant and depends only on $L/W$ and the boundary condition. It is also shown that once disorder is introduced, $\langle\sigma\rangle$ monotonically increases with increasing $L$. Hence, the system becomes a perfect metal in the limit of $L \to \infty$ except at the clean limit, which should be identified as an unstable fixed point. Although the scaling curve of $\langle\sigma\rangle$ strongly depends on $L/W$ and the boundary condition near the unstable fixed point, it becomes almost independent of them with increasing $\langle\sigma\rangle$, implying that it asymptotically obeys a universal law.