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

TL;DR

This paper explores the complex interplay between first-order and second-order topological phases on the honeycomb lattice, revealing a rich phase diagram with novel topological states and corner modes at various fillings.

Contribution

It provides a comprehensive analysis of the competition between first- and second-order topology, identifying new topological phases and real-space corner states on the honeycomb lattice.

Findings

Discovery of twelve distinct topological phases at half-filling.

Identification of a novel $ ext{Z}_6$ topological phase.

Observation of corner states at multiple fillings.

Abstract

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 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.