Fragile topology for six-fold rotation symmetry indicated by the concentric Wilson loop spectrum

Abstract: We investigate topological phase transitions for the Haldane and Kane-Mele model in a lattice with $p6$ symmetry, which consists of triangles and hexagons arranged in a two-dimensional geometry. For the Haldane model, which breaks time-reversal symmetry, we calculate the Chern number using a multi-band non-Abelian Wilson loop formalism. By varying the hopping parameters in the triangles and hexagons independently, a large variety of topological phases emerge. In the presence of a next-next-nearest neighbor hopping, the phase diagram becomes even richer, with regions exhibiting high Chern numbers. Then, we consider the Kane-Mele model, for which time-reversal symmetry is preserved, and calculate the number of $π$-crossings in the Concentric Wilson Loop Spectrum (CWLS). This method is appropriate to determine the topological invariant for systems hosting time-reversal and rotational symmetry, but lacking all other symmetries. According to a classification based on $K$-theory, the CWLS invariant reveals topological properties even when more conventional invariants fail to detect them. The formalism was previously successfully applied to systems with 3- and 4-fold symmetry. Here, we surprisingly find that for the 6-fold-symmetry model investigated, the topology identified by this invariant is fragile, therefore questioning the claim that this should be the strong invariant missing in a complete classification of topological insulators.