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

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.

💡 Research Summary

In this work the authors present a comprehensive tight‑binding study of edge‑state physics in the two‑dimensional kagome lattice, focusing on three key ingredients: lattice termination, spin‑orbit coupling (SOC), and magnetic order. Starting from a single‑orbital nearest‑neighbour hopping model, they first map out the bulk band structure, which consists of two dispersive bands featuring Dirac points at the K/K′ corners and a completely flat band that gives rise to a van‑Hove singularity. By constructing slab geometries with four distinct terminations—zigzag, armchair, “cove”, and flat—they demonstrate that the existence, dispersion, and spatial localisation of edge modes are extremely sensitive to the boundary geometry. Armchair and zigzag edges host two in‑gap edge states (one connecting the Dirac points and another near the flat band), while the flat termination completely suppresses edge modes. The momentum‑resolved local density of states (LDOS) reveals that edge states preferentially occupy sites with two or four nearest neighbours, leading to a finite orbital angular momentum that could be exploited in orbitronics.

The authors then introduce intrinsic Kane‑Mele SOC. With a moderate coupling (λKM≈0.15 t) the bulk spectrum opens a global gap and the Wilson‑loop calculation yields a Z₂ invariant of 1, confirming a quantum spin‑Hall (QSH) phase. In this regime helical edge states appear irrespective of the termination, and their spin is locked to the propagation direction, illustrating the robustness of the Z₂ topology against edge geometry.

Next, they add a Zeeman field together with Rashba SOC. The Zeeman term splits the spin bands, while Rashba SOC lifts the remaining degeneracies, generating Chern insulating phases. By varying the Zeeman strength and Rashba amplitude they obtain Chern numbers C = ±1, ±2, etc., and the number of chiral edge modes in the slab exactly matches |C|, confirming bulk‑boundary correspondence for the anomalous Hall regime. Increasing Rashba coupling widens the topological gap, suggesting a route to more stable chiral channels.

Finally, the study explores non‑coplanar magnetic textures (umbrella‑type ordering) that produce a finite scalar spin chirality χ = ⟨S_i·(S_j×S_k)⟩. This chirality acts as an emergent magnetic flux, opening additional gaps and stabilising multiple Chern phases (e.g., C = ±3). When intrinsic SOC is also present, the gaps become substantially larger, and the edge spectrum exhibits a rich spin texture combining helical and chiral characteristics.

Overall, the paper establishes four central conclusions: (i) edge‑state existence in kagome lattices is dictated by termination; (ii) intrinsic SOC yields a termination‑independent Z₂ QSH phase; (iii) Zeeman + Rashba SOC drives Chern insulating behavior with a direct link between Chern number and chiral edge count; (iv) non‑coplanar magnetic order introduces scalar chirality, generating multiple Chern phases and enhancing topological gaps. These insights provide concrete design principles for engineering kagome‑based 2D materials with controllable edge conduction, spin‑polarised transport, and robust topological functionalities.