Non-commutative odd Chern numbers and topological phases of disordered chiral systems

An index theorem for higher Chern characters of odd Fredholm modules over crossed product algebras is proved, together with a local formula for the associated cyclic cocycle. The result generalizes the classic Noether–Gohberg–Krein index theorem, which in its simplest form states that the winding number of a complex-valued function over the circle is equal to the index of the associated Toeplitz operator. When applied to the non-commutative Brillouin zone, this generalization allows to define topological invariants for all 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 a region of Anderson localized spectrum.