Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice

Like all topologically nontrivial bands, the recently discovered fragile topological states cannot be described with symmetric localized basis functions. Unlike the well-understood ``stable'' topological phases, fragile ones have a localized description in the presence of additional trivial orbitals. Here, the authors explore the properties of fragile topological bands, focusing on a family of models on a triangular lattice. The authors show how nonabelian Berry phases can be used to diagnose fragile topology. They also explore how fragile topology manifests itself in the entanglement spectrum.