Quantum Phase diagrams and transitions for Chern topological insulators

Abstract: Topological invariants such as Chern classes are by now a standard way to classify topological phases. Introducing and varying parameters in such systems leads to phase diagrams, where the Chern classes may jump when crossing a critical locus. These systems appear naturally when considering slicing of higher dimensional systems or when considering systems with parameters. As the Chern classes are topological invariants, they can only change if the "topology breaks down". We give a precise mathematical formulation of this phenomenon and show that synthetically any phase diagram of Chern topological phases can be designed and realized by a physical system, using covering, aka. winding maps. Here we provide explicit families realizing arbitrary Chern jumps. The critical locus of these maps is described by the classical rose curves. These realize the lower bound on the number of Dirac points necessary obtained from viewing them as local charges. We treat several concrete models and show that they have the predicted generic behavior. In particular, we focus on different types of lattices and tight-binding models, and show that effective winding maps, and thus higher Chern numbers, can be achieved using k-th nearest neighbors. We give explicit formulas for a family of 2D lattices using imaginary quadratic field extensions and their norms. Our study includes the square, triangular, honeycomb and Kagome lattices.