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

We consider  a wide class of  2D tight - binding models of solid state physics. These models are, in the most general case, non - homogeneous. Lattice Dirac operator Q (i.e. the inverse two point Green function) of the considered systems contains a rather general dependence on Matsubara frequency that is not reduced to the Hamiltonian case Q = i w - H with a Hamiltonian H. The topological invariant  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 HC^3. According to a set of local index theorems  the values of N_3  can be shown to be integer for a wide but limited class of  the lattice systems. In the Hamiltonian case our construction is reduced to that of Bellissard and co - authors. We concentrate on systems with a spectral gap, with the chemical potential inside the gap and an invertible Dirac operator Q throughout the Matsubara-frequency domain. The mobility-gap case is not considered here and will be addressed in forthcoming publications.

Following https://arxiv.org/abs/2606.08868