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