Existence of zero-energy impurity states in different classes of topological insulators and superconductors and their relation to topological phase transitions

We consider the effects of impurities on topological insulators and superconductors. We start by identifying the general conditions under which the eigenenergies of an arbitrary Hamiltonian H belonging to one of the Altland-Zirnbauer symmetry classes undergo a robust zero energy crossing as a function of an external parameter which can be, for example, the impurity strength. We define a generalized root of \det H, and use it to predict or rule out robust zero-energy crossings in all symmetry classes. We complement this result with an analysis based on almost degenerate perturbation theory, which allows a derivation of the asymptotic low-energy behavior of the ensemble averaged density of states $\rho \sim E^\alpha$ for all symmetry classes, and makes it transparent that the exponent \alpha\ does not depend on the choice of the random matrix ensemble. Finally, we show that a lattice of impurities can drive a topologically trivial system into a nontrivial phase, and in particular we demonstrate that impurity bands carrying extremely large Chern numbers can appear in different symmetry classes of two-dimensional topological insulators and superconductors. We use the generalized root of \det H(k) to reveal a spiderweblike momentum space structure of the energy gap closings that separate the topologically distinct phases in p_x + i p_y superconductors in the presence of an impurity lattice.