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