The non-commutative \(n\)th-Chern number \((n\geq 1)\) (Q2871964)

The authors work with the projectors onto the occupied electron states of typical disordered \(2n\)-dimensional lattice-models. These models are discussed starting from the simplest case of translation invariant ones and ending with the general class of homogeneous lattice systems, which includes the disordered quantum lattice-models under uniform magnetic fields. Several representations of the \(n\)th-Chern number are given. Moreover the topological properties of the \(n\)th-Chern number are established. Namely, its quantization and homotopy invariance, together with the optimal conditions when these happen are shown. The relevance of the result to the field of strongly disordered topological insulators is discussed.