Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond

Within topological band theory, stable, fragile, and recently discovered delicate topological insulators are distinguished based on their stability against addition of trivial degrees of freedom. In this manuscript, the authors study delicate topological insulators protected by crystalline symmetries. They demonstrate that sharp boundaries of these insulators possess conducting states, while states localized at nonsharp boundaries are generically insulating but have anomalous symmetry eigenvalues. These boundary signatures are qualitatively unique among the known topological insulators.