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

Understand

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.