Topology of honeycomb nanoribbons revisited

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.

Figures (9)

• Figure 1: The bulk spectra of one-dimensional 'zigzag' nanoribbons. The width of the considered nanoribbons is (a) 7 hexagons, (b) 3 hexagons, and (c) 1 hexagon. As the width decreases, a gap opens due to the real-space hybridization of the chiral edge modes. This hybridization gap increases in size as the width of the nanoribbon decreases. Here, $M = -0.5$, $t_2 = 0.3$, and $a$ is the lattice constant.
• Figure 2: The energy gap, Zak phase, and OBC spectrum of zigzag Haldane nanoribbons as a function of $M$. (a)-(e), the bulk energy gap of the nanoribbons, (f)-(j), the Zak phase in the bulk gap, and (k)-(o) the OBC spectrum of a 500-unit-cell-long nanoribbon. These properties are shown for nanoribbons of different widths: (a, f, k) 1-hexagon, (b, g, l) 2-hexagon, (c, h, m) 3-hexagon, (d, i, n) 4-hexagon, and (e, j, o) 5-hexagon wide nanoribbons. Here, $t_2=0.3t$.
• Figure 3: The energy gap, Zak phase, and OBC spectrum of armchair Haldane nanoribbons as a function of $M$. (a)-(e), the bulk energy gap of the nanoribbons, (f)-(j), the Zak phase in the corresponding gap, and (k)-(0) the OBC spectrum of a 500-unit-cell-long nanoribbon. Note that these are not dispersing modes, but the evolution of OBC spectra with $M$. These properties are shown for nanoribbons of different widths: (a, f, k) 1-hexagon, (b, g, l) 1.5-hexagon, (c, h, m) 2-hexagon, (d, i, n) 2.5-hexagon, and (e, j, o) 3-hexagon wide nanoribbons.
• Figure 4: The Zak phase and open spectra corresponding to an odd (top) and even (bottom) zigzag nanoribbons with different terminations. In (a)-(d), the Zak phase [(a), (c)] and the spectrum of a 500-unit-cell-long nanoribbon [(b), (d)] are shown for a rectangular zigzag nanoribbon. Analogously, (e)-(h) show the results on a modified rectangular zigzag nanoribbon, (i)-(l) a rhombic nanoribbon, and (m)-(p) a modified rhombic nanoribbon. Quantization of the Zak phase is only observed in (a) and (e) because these are the only systems with a chiral symmetry.
• Figure 5: The complex NNN hopping phase $\phi$ dependence of the hybridization gap and the Zak phase as a function of staggered mass $M$ of odd [(a),(c), and (e)] and even [(b), (d), and (f)] zigzag nanoribbons, 1-hexagon- and 2-hexagons wide, respectively. In (a) and (b), the hybridization gap size is shown for 5 values of $\phi$, as indicated by the color bar. In both cases, the hybridization gap decreases in size as the complex hoppings are rotated onto the real axis, and vanishes when the NNN hopping has become fully real. Furthermore, the gap opening conditions become more constrained, and as the NNN hopping becomes real, the gap opens for smaller ranges of the staggered mass. In (c) and (d), we see that the chiral symmetry of the $\phi=\pi/2$ case is lost, because the Zak phase loses its quantization. Open spectra for $\phi=3\pi/8$ are shown in (e) and (f), where unpinned edge modes are observed.