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

TL;DR

This paper investigates the finite-size scaling behavior of conductivity in 2D Dirac electrons on disordered topological insulator surfaces, revealing a universal law governing the transition to perfect metallic conduction.

Contribution

It provides a numerical analysis of conductivity scaling at the Dirac point, identifying an unstable fixed point and universal behavior in disordered topological insulator surfaces.

Findings

Conductivity is minimized at the clean limit and is scale-invariant.

Disorder causes conductivity to increase with system size, leading to a perfect metal.

Scaling curves become universal as conductivity increases.

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.