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′ 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.

中文翻译：

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二维高阶拓扑金属


我们研究了具有交错本征自旋轨道耦合和面内塞曼场的石墨烯的能带结构和能级。我们的研究表明，交错的本征自旋轨道耦合在 K 和 K′ 谷处诱导体带交叉，并在锯齿形边界处产生反螺旋边缘状态，从而产生拓扑金属相。量子化传输系数证实了这些反螺旋边缘状态的存在。此外，无论方向如何，面内塞曼场都会在反螺旋边缘状态中打开一个间隙，同时保持体带闭合，从而导致具有拐角状态的高阶拓扑金属。我们还验证了具有锯齿形边界的纳米薄片中存在这些角态，并通过连续低能模型分析确认了具有交叉带的金属相。