Disorder-induced density of states on the surface of a spherical topological insulator

We consider a topological insulator of spherical geometry and numerically investigate the influence of disorder on the density of surface states. To the clean Hamiltonian we add a surface disorder potential of the most general Hermitian form, $V = V^0(\theta,\phi) + {\bf V}(\theta,\phi) \cdot {\bf \sigma}$. We expand these four disorder functions in spherical harmonics and draw the expansion coefficients randomly from a 4D Gaussian distribution. For each disorder instantiation, we solve for the energy spectrum via exact diagonalization. Then we compute the disorder-averaged density of states by averaging over 200,000 different instantiations. Disorder broadens the Landau-level delta functions of the clean density of states into peaks that decay and merge together. Increasing disorder strength pushes states closer to zero energy, resulting in a low-energy density of states that becomes nonzero for sufficient disorder, typically approaching an energy-independent saturation value. But for purely spin-dependent disorder with ${\bf V}$ either entirely out-of-surface or entirely in-surface, we identify intriguing disorder-induced features in the vicinity of the Dirac point. These are explained in terms of the breaking (or not) of two chiral symmetries of the clean Hamiltonian.