Non-commutative odd Chern numbers and topological phases of disordered   chiral systems
  An odd index theorem for higher odd Chern characters of crossed product algebras is proved. It generalizes the Noether-Gohberg-Krein index theorem. Furthermore, a local formula for the associated cyclic cocycle is provided. When applied to the non-commutative Brillouin zone, this allows to define topological invariants for condensed matter phases from the chiral unitary (or AIII-symmetry) class in the presence of strong disorder and magnetic fields whenever the Fermi level lies in region of Anderson localization. 
Emil Prodan, Hermann Schulz-Baldes
math-ph cond-mat.dis-nn math.MP math.OA