The topological invariants of rotationally symmetric crystals

Abstract: Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with two-fold and three-fold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges.