Competition of First-Order and Second-Order Topology on the Honeycomb Lattice

Abstract

We investigate both first-order topology, as realized through Haldane's model, and second-order topology, implemented through an additional Kekulé-distortion, on the honeycomb lattice. The interplay and competition of both terms result in a phase diagram at half-filling which contains twelve 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 novel ${\mathbb{Z}}_6$ topological phase, but also coupled kagome-lattice Chern insulators. Furthermore, we explore the insulating phases at lower fillings, and find again first-order 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.

19 pages, 14 figures

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• Kekulé valence bond order in the honeycomb lattice optical Su-Schrieffer-Heeger Model and its relevance to Graphene
• $Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder
• Extended Haldane model -- a modern gateway to topological insulators
• Hybrid Higher-Order Skin Topological Modes in the Two-Dimensional Su-Schrieffer-Heeger Model with Nonreciprocal Hoppings
• Single $\pi$-flux hosting topological defect modes in bilayer acoustic metamaterials