Quantum Spin Hall Phase in the Truncated Trihexagonal Lattice: A Topological Archimedean Structure

TL;DR

The paper develops a Python-based tight-binding framework to study all eight pure Archimedean lattices realized as 2D carbon networks, focusing on band topology, edge states, and spin transport. By including $s$ and $p$ orbitals, NN and NNN hopping, and proximity-induced SOC with $\lambda_{\mathrm{SOC}} = 1\,\mathrm{eV}$, it computes bulk and ribbon band structures, $ ext{Z}_2$ invariants, and intrinsic spin Hall conductivities via the Kubo formalism. A key finding is that several lattices, notably the truncated trihexagonal, host robust topological edge states and a quantized spin Hall response ($\sigma_{xy}^z \approx e/\pi$) within a global bulk gap, corresponding to a spin Chern number $C_s = 2$. Flat bands persist across the Brillouin zone in some lattices, suggesting potential for strongly correlated physics, while other Archimedean lattices show trivial topology or Dirac-like degeneracies; the work provides open-source tools and CIF data to enable further exploration and experimental realization attempts, including possible SOC enhancements via heavier elements or TMDC proximity.

Abstract

Archimedean lattices constitute a unique family of two-dimensional tilings formed from regular polygons arranged with uniform vertex configurations. While the kagome and snub square lattices, the simplest members of the Archimedean lattice family, have been extensively investigated -- the former as a paradigmatic system for geometric frustration and nontrivial band topology, and the latter primarily as a quasicrystal approximant -- the broader family remains largely unexplored in terms of electronic and topological properties. In this work, we present a systematic Python-based tight-binding study of all eight pure Archimedean lattices, modeled as two-dimensional carbon-based networks serving as a proof-of-principle system. We analyze their band structures, investigate topological edge states arising from unconventional nanoribbon geometries, and evaluate $\mathbb{Z}_2$ invariants as well as intrinsic spin Hall conductivities using the Kubo formalism. Our results reveal that several Archimedean lattices, such as the truncated hexagonal and truncated trihexagonal lattices, host nearly dispersionless flat bands extending across the Brillouin zone, which remain robust even in the presence of next-nearest-neighbor hopping and strong spin-orbit coupling. In particular, the truncated trihexagonal lattice supports topologically protected, highly spin-polarized edge states across multiple ribbon geometries. These states are stable against defects and spin-flip scattering, and give rise to quantized spin Hall currents.

Figures (8)

• Figure 1: The unit cell of the snub square lattice
• Figure 2: Band structure of the kagome lattice, including NN hopping terms (left) and both NN and NNN (right) hopping terms, the $p_z$ projected bands are highlighted in red
• Figure 3: The band structure of the kagome lattice in an interval between $-8 \ \mathrm{eV}$ and $8 \ \mathrm{eV}$. The bands are color-coded according to the expectation value $\langle S_z \rangle$. Different interactions are discussed: (a) only Zeeman interaction ($\lambda_{\text{ZM}} = 0.2 \ \mathrm{eV}$) (b) both Zeeman ($\lambda_{\text{ZM}} = 0.2 \ \mathrm{eV}$) and SOC ($\lambda_{\text{SOC}} = 1.0 \ \mathrm{eV}$) (c) only SOC ($\lambda_{\text{SOC}} = 1.0 \ \mathrm{eV}$). In the latter case, the bands remain fully degenerate because time-reversal and inversion symmetry are preserved. Only the band with $\langle S_z \rangle \approx 1$ is visible.
• Figure 4: Band structures of (a) the kagome (b) the truncated hexagonal and (c) the truncated trihexagonal lattices with and without SOC. Both NN and NNN hopping terms are included. The strength of SOC, if present, is $1 \ \mathrm{eV}$.
• Figure 5: (a) From left to right: zigzag-armchair, armchair-armchair, arc armchair ribbons of the truncated trihexagonal lattice (b) Bulk and ribbon band structures of the truncated trihexagonal lattice. Both NN and NNN hopping terms, as well as SOC with $\lambda_{\text{SOC}} = 1 \ \mathrm{eV}$ are included. The periodicity is set in the horizontal direction for all ribbons. Here, all band structure calculations are performed on ribbons with a width of 12 unit cells in the non periodic direction and 1 unit cell in the periodic direction.