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

TL;DR

This paper introduces a coarse geometric framework for topological phases that incorporates disorder, providing a natural setting for bulk-boundary correspondence and enabling efficient numerical invariant calculations.

Contribution

It develops a novel coarse geometric approach to topological invariants that applies to disordered materials and demonstrates its effectiveness through numerical simulations.

Findings

Reproduces known phase diagrams in disordered systems

Provides a tractable numerical method for topological invariants

Analyzes topological phase evolution under disorder

Abstract

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.