Competition of first-order and second-order topology on the honeycomb lattice

We investigate both first-order topology, as realized through Haldane's model, and second-order topology, implemented through an additional Kekul\'e distortion, on the honeycomb lattice. The interplay and competition of both terms result in a phase diagram at half-filling which contains 12 distinct phases. All phases can be characterized by the first Chern number or by a quantized ${\mathbb{Z}}_Q$ Berry phase. Highlights include phases with high Chern numbers, a ${\mathbb{Z}}₆$ topological phase, but also coupled kagome-lattice Chern insulators. Furthermore, we explore the insulating phases at lower fillings and find again first- and second-order topological phases. Finally, we identify real-space structures which feature corner states not only at half but also at third and sixth fillings, in agreement with the quantized ${\mathbb{Z}}_Q$ Berry phases.