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We consider the effects of impurities on topological insulators andsuperconductors. We start by identifying the general conditions under which theeigenenergies of an arbitrary Hamiltonian H belonging to one of theAltland-Zirnbauer symmetry classes undergo a robust zero energy crossing as afunction of an external parameter which can be, for example, the impuritystrength. We define a generalized root of \det H, and use it to predict or ruleout robust zero-energy crossings in all symmetry classes. We complement thisresult with an analysis based on almost degenerate perturbation theory, whichallows a derivation of the asymptotic low-energy behavior of the ensembleaveraged density of states $\rho \sim E^\alpha$ for all symmetry classes, andmakes it transparent that the exponent \alpha\ does not depend on the choice ofthe random matrix ensemble. Finally, we show that a lattice of impurities candrive a topologically trivial system into a nontrivial phase, and in particularwe demonstrate that impurity bands carrying extremely large Chern numbers canappear in different symmetry classes of two-dimensional topological insulatorsand superconductors. We use the generalized root of \det H(k) to reveal aspiderweblike momentum space structure of the energy gap closings that separatethe topologically distinct phases in p_x + i p_y superconductors in thepresence of an impurity lattice.

physical review. b./physical review. b

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