Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder

Abstract

We compute the multifractal spectra associated to local density of states (LDOS) fluctuations due to weak quenched disorder for a single Dirac fermion in two spatial dimensions. Our results are relevant to the surfaces of ${\mathbb{Z}}_{2}$ topological insulators such as ${\text{Bi}}_{2}{\text{Se}}_{3}$ and ${\text{Bi}}_{2}{\text{Te}}_{3}$, where LDOS modulations can be directly probed via scanning tunneling microscopy. We find a qualitative difference in spectra obtained for magnetic versus nonmagnetic disorder. Randomly polarized magnetic impurities induce quadratic multifractality at first order in the impurity density; by contrast, no operator exhibits multifractal scaling at this order for a nonmagnetic impurity profile. For the time-reversal invariant case, we compute the first nontrivial multifractal correction, which appears at two loops (impurity density squared). We discuss spectral enhancement approaching the Dirac point due to renormalization, and we survey known results for the opposite limit of strong disorder.