The Noncommutative Index Theorem and the Periodic Table for Disordered Topological Insulators and Superconductors

We study a wide class of free-fermion systems on a hypercubic lattice in spatial dimensions d ≥ 1. When the system is insulating, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analogue of the Atiyah-Singer index theorem. The noncommutative index defined in terms of a pair of projections gives a precise formula for the topological invariant in each symmetry class in any dimension (d ≥ 1). The analysis of the index reproduces Bott periodicity and possible values of topological invariants in the classification table of topological insulators and superconductors. We also prove that the indices are robust against perturbations that do not break the symmetry of the unperturbed Hamiltonian, provided the Fermi level lies in a spectral gap or a mobility gap.