Two-dimensional higher-order topological metals

We investigate the energy band structure and energy levels of graphene with staggered intrinsic spin-orbit coupling and in-plane Zeeman fields. Our study demonstrates that staggered intrinsic spin-orbit coupling induces bulk band crossover at the $K$ and ${K}^{\ensuremath{'}}$ valleys and generates antihelical edge states at the zigzag boundaries, resulting in topological metallic phases. Quantized transport coefficients confirm the existence of these antihelical edge states. Furthermore, an in-plane Zeeman field, regardless of orientation, opens a gap in the antihelical edge states while preserving bulk band closure, leading to higher-order topological metals with corner states. We also validate the presence of these corner states in nanoflakes with zigzag boundaries and confirm the metallic phases with crossed bands through a continuum low-energy model analysis.