Coarse geometric approach to topological phases: Invariants from real-space representations

We show that topological phases include disordered materials if the underlying invariant is interpreted as originating from coarse geometry. This coarse geometric framework, grounded in physical principles, offers a natural setting for the bulk-boundary correspondence, reproduces physical knowledge, and leads to an efficient and tractable numerical approach for calculating invariants. As a showcase, we give a detailed discussion of the framework for three-dimensional systems with time-reversal symmetry. We numerically reproduce the known disorder-free phase diagram of a tunable, effective tight-binding model and analyze the evolution of the topological phase under disorder.