Mirror winding number and helical edge modes in honeycomb lattice with hopping-energy texture

TL;DR

This paper explores topological phases in honeycomb lattices with textured hopping energies, revealing mirror winding numbers and helical edge modes, and suggests edge decoration as a method to induce topological states.

Contribution

It introduces the concept of mirror winding numbers in honeycomb lattices with hopping textures and demonstrates how edge decoration can induce topological phases.

Findings

Identification of topological phases characterized by mirror winding numbers.

Analytic solutions for zero-energy edge modes classified by mirror symmetry.

Edge decoration can induce topological helical edge states.

Abstract

We illustrate possible topological phases in honeycomb lattice with textures in electron hopping energy between nearest-neighboring sites and show that they are characterized by the mirror winding number intimately related to the chiral (or sublattice) symmetry. Analytic wave functions of zero-energy edge modes in ribbon geometry are provided, which are classified into even and odd sectors with respect to the mirror operation with the mirror plane perpendicular to the edge, and evolve into the topological helical edge states at finite momenta. Intriguingly our results demonstrate that in order to achieve the topological phase one can decorate the edge in a way adaptive to the bulk hopping texture. This paves a new way to tailoring graphene in the topological point of view.