Abstract: For perfectly periodic crystals, the topological invariants, the linear response coefficients and the correlation functions in general are expressed using the differential geometry over the classic Brillouin torus. Using an equivalence between a manifold and its algebra of functions, one can extend the notion of the Brillouin torus to that of a non-commutative Brillouin torus. Together with a properly defined non-commutative differential geometry (NCG), the non-commutative Brillouin torus gives a natural framework to express and compute the topological invariants, the linear response coefficients and other correlation functions for disordered systems in the presence of magnetic fields.
In this talk, I will present first some analytical results concerning the quantization and invariance of topological numbers in the presence of strong disorder. I will try to highlight the critical conceptual difficulties introduced by the strong disorder and then to showcase the elegant solutions provided by NCG. In the second part I will demonstrate that the non-commutative differential formalism can be used to construct extremely efficient numerical algorithms to compute generic correlation functions for disordered systems in the presence of magnetic fields. Using such algorithms for the Chern and the spin-Chern number, we have mapped the phase diagrams of various topological insulators as functions of disorder strength, Fermi level and other system parameters. Here I will focus on one particular study, where we were able to demonstrate that the so called Topological Anderson Insulating phase is connected to the Quantum spin-Hall phase and, as such, to show that they are one and the same phase of matter. Furthermore, using the non-commutative Kubo formula, we have completed several studies of the electron-transport in disordered topological insulators, with and without magnetic fields. For a model Quantum spin-Hall insulator, I will present maps of the conductivity tensor as function of disorder, temperature and Fermi level. These maps give a direct evidence, via the temperature behavior of the transport coefficients, for the existence of a metallic (and not just extended) phase between the topological and trivial phases. I will also demonstrate a universal critical behavior of the transport coefficients at the trivial-Chern insulator transition. Specifically, I will show quantitatively that this transition is similar in all respects to the plateau-insulator transition in the integer Quantum Hall Effect.