Kagome edge states under lattice termination, spin-orbit coupling, and magnetic order

TL;DR

This work addresses how edge states in a 2D kagome lattice respond to lattice termination, spin–orbit coupling, and magnetic order using a tight-binding framework. It shows that pristine edges are highly termination-dependent, while Kane–Mele SOC opens bulk gaps and yields a robust $\mathbb{Z}_2$ quantum spin Hall phase with termination-insensitive helical edge modes. When TRS is broken by ferromagnetism (Zeeman field with Rashba SOC) or non-coplanar magnetic order, quantum anomalous Hall phases with tunable Chern numbers $\mathcal{C}$ emerge, with Kane–Mele SOC providing additional control over gap sizes and edge dispersions. Overall, the kagome lattice is demonstrated as a versatile platform for engineering tunable edge states and topological phases, with potential implications for surface engineering in kagome materials.

Abstract

We study the edge state properties of a two-dimensional kagome lattice using a tight-binding approach, focusing on the role of lattice termination, spin-orbit coupling, and magnetic order. In the pristine limit, we show that the existence of localized edge states is highly sensitive to boundary geometry, with certain terminations completely suppressing edge modes. Kane-Mele spin-orbit coupling opens a bulk gap and stabilizes topologically protected helical edge states, yielding a robust $\mathbb{Z}_2$ insulating phase that is insensitive to termination details. In contrast, the combined effect of a Zeeman field and Rashba spin-orbit coupling drives the system into Chern insulating phases, with Chern numbers consistent with the number of chiral edge modes. We further demonstrate that non-coplanar magnetic textures generate multiple Chern phases through finite scalar spin chirality, with Kane-Mele coupling strongly tuning the topological gaps. Our results provide important insights into the tunability of edge states in the kagome lattice, which can be key to designing materials with novel electronic properties and topological phases.

Figures (9)

• Figure 1: (a) Kagome lattice, schematically highlighting the four edge types arising from termination by a line cut. Solid arrows indicate nearest $\mathbf{d}_{1,2,3}$ and next-nearest $\mathbf{d}^{\prime}_{1,2,3}$ neighbor vectors, while site color-coding refers to the three sublattice atoms of the kagome lattice. (b) Band structure (left) and density of states (DOS) (right) in the kagome lattice, with features of Dirac point (DP), van Hove singularity (VHS), and flat band (FB) appearing at different fillings. (c) Full (blue hexagon) and projected slab (blue line) Brillouin zones of the kagome lattice. In contrast to the full Brillouin zone, the high symmetry points in the slab are written with bar. Yellow cones denote the Dirac points.
• Figure 2: Kagome slab band structure for different lattice terminations (left), with energy levels at a specific momentum (red vertical line) to pinpoint the edge states (middle), edge states are highlighted by icons, and we present the momentum-resolved site LDOS on real-space lattice (right). Parameters are $\mu/t=0$ and $t=1$.
• Figure 3: (a) Bulk kagome bands for the QSH phase and the slab band structure for (b) armchair-flat and (c) zigzag-cove type terminations. In all bandstructure plots, we take $\lambda_{\text{KM}}/t=0.15$ and $\mu/t=0$. Colors show expectation of the position operator. (d) Eigenvalues of the Wilson loop plotted as a function of $k_y$ for two different fillings, as indicated in (a). Eigenvalues of both fillings $\langle n \rangle$ cross the reference line (blue line) once (odd number of times), indicating the $\mathbb{Z}_2$ nature.
• Figure 4: (a) Bulk kagome bands for the QAH phase without (red) and with RSOC ($\lambda_{R}/t=0.2$) (blue dashed) under constant Zeeman field ($h_z/t=0.6$). Slab band structures with Zeeman and RSOC for (b) armchair-flat and (c) zigzag-cove terminations, with colors showing the expectation value of the position operator. (d) Anomalous Hall conductivity (in units of $e^2/h$) as a function of chemical potential. Circular cyan icons highlight the location of the VHS points. Insets show the Berry curvature maps in the two topological gaps, with icons also indicated in (a). Enclosed cyan line denote the BZ. (e) Fermi contours plotted at ($\mu/t=0.6$) and around $\mu/t=(0.3,0.9$) the VHS 4 in (d).
• Figure 5: (a) Bulk kagome bands for the QAH phase without (red) and with KMSOC ($\lambda_{\textrm{KM}}/t=0.15$) (blue dashed) under constant Rashba SOC ($\lambda_{R}/t=0.2$) and Zeeman field ($h_z/t=0.6$). Slab band structures with Zeeman, RSOC, and KMSOC for (b) armchair-flat and (c) zigzag-cove terminations, with colors showing the expectation of the position operator. (d) Phase diagram capturing the evolution of the anomalous Hall conductivity $\sigma_{xy}$ (in terms of $e^2/h$) as a function of chemical potential $\mu$ and Kane--Mele coupling $\lambda_{\mathrm{KM}}$. Enclosed regions indicate quantized QAH phases with finite Chern number $\mathcal{C}$. In all plots, triangle and star icons refer to different topological gaps with $\mathcal{C}=2$ and $\mathcal{C}=-1$, respectively.