Topology of honeycomb nanoribbons revisited

TL;DR

This paper provides a detailed topological analysis of honeycomb nanoribbons, examining how termination, chiral symmetry, and complex hopping influence end states within the Haldane and Kane-Mele models.

Contribution

It systematically investigates the topological properties of nanoribbons using the multiband Zak phase, clarifying the conditions for end state existence and robustness.

Findings

Zak phase quantization depends on ribbon termination

End states are governed by termination-dependent topological invariants

Complex hopping phase affects chiral symmetry and end-state energies

Abstract

We present an in-depth study of end states in honeycomb nanoribbons, focusing on the interplay between nanoribbon termination, chiral symmetry, and complex next-nearest-neighbor hopping in the framework of the Haldane model. Although previous work has identified zero-dimensional end states in such systems, this analysis is incomplete. Here, we systematically investigate zigzag and armchair nanoribbons of various widths, using the multiband Zak phase to characterize the topological properties of the occupied bands. We show that the Zak phase is quantized only for certain ribbon terminations, and we elucidate how this termination dependence governs the existence and robustness of end states. Furthermore, we explore the effect of varying the complex next-nearest-neighbor hopping phase, demonstrating the breakdown of chiral symmetry, the evolution of the bulk gap, and the resulting depinning of end-state energies. Finally, we place our findings in the context of previous studies and discuss connections to the Kane-Mele model, including the role of Rashba spin-orbit coupling. Our work provides a more detailed analysis of topological end states in nanoribbons described by the Haldane and Kane-Mele models and offers a framework for their characterization in related systems.