Topological invariant responsible for the integer QHE and non-commutative geometry

Abstract

We consider a wide class of $2D$ tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. The topological invariant ${\mathcal{N}}_3$ responsible for the quantization of the Hall conductivity, for the specific case of the integer quantum Hall effect in $2D$, is expressed through the Wigner transformation of the two-point electron Matsubara Green function. We express this invariant as a pairing of the element of the $K^{-1}$ group (generated by the Green function) with the specific element of the cyclic cohomology group $H{C}^3$. According to a set of local index theorems the values of ${\mathcal{N}}_3$ can be shown to be integer for a limited class of tight - binding models.

Latex, 65 pages, 1 figure