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

We consider a topological insulator (TI) 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 four-dimensional, zero-mean gaussian distribution. Different strengths and classes of disorder are realized by specifying the $4 \times 4$ covariance matrix. For each instantiation of the disorder, we solve for the energy spectrum via exact diagonalization. Then we compute the disorder-averaged density of states, $\rho(E)$, 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. If the spin-dependent term is dominant, these peaks split due to the breaking of the degeneracy between time-reversed partner states. Increasing disorder strength pushes states closer and closer to zero energy (the Dirac point), resulting in a low-energy density of states that becomes nonzero for sufficient disorder, typically approaching an energy-independent saturation value, for most classes of disorder. 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. In the out-of-surface case, a new peak emerges at zero energy. In the in-surface case, we see a symmetry-protected zero at zero energy, with $\rho(E)$ increasing linearly toward nonzero-energy peaks. These striking features are explained in terms of the breaking (or not) of two chiral symmetries of the clean Hamiltonian.