Edge states and the valley Hall effect
Edge states and the valley Hall effect
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DOI：
10.1016/j.aim.2020.107142
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发表时间：
2019-10
影响因子：
1.7
通讯作者：
A. Drouot;M. Weinstein
中科院分区：
文献类型：
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作者：
A. Drouot;M. Weinstein
研究了二维连续保能周期介质中沿线缺陷（边缘）的能量传播。无扰动介质（体）由蜂窝薛定谔算子建模，它相对于三角形晶格是周期性的，在奇偶性P和复共轭c下不变。蜂窝算子在其能带结构中有狄拉克点：两个色散面在能级E D[25]，[27]上圆锥接触。打破P或C的周期扰动在能量E d的基本谱中打开了一个缺口。这样的算子模拟了能量E d附近的绝缘子。我们的边缘算子是体的一个小扰动，并模拟了不同周期，P或C破缺扰动之间的过渡（通过畴壁）。边缘算子允许能量沿线缺陷传输。相关的能量通道称为边缘态。它们是底层波动方程的时谐解，在线缺陷附近局部化并沿线缺陷传播。由于其显著的稳定性，它们引起了极大的科学兴趣，并且是拓扑绝缘体的关键特性。我们完整地描述了在E d附近的体谱间隙内的边缘态谱。我们分析的中心是E d附近能量的边缘算子解析的展开。边缘状态特征值是解的极点，它从Dirac点分叉。相应的特征态具有[23]中识别的多尺度结构。我们将之前关于之字形边[14]的工作扩展到所有有理边。我们阐明了对称破缺的类型和边的方向在边态形成中的作用。我们通过一种新的直接和透明的策略证明了可解决的扩展。我们的结果也为[22]、[38]的数值观测提供了严格的解释；另见[42]的光子实验研究。最后，我们讨论了谷霍尔效应的影响，它涉及蜂窝结构中的量子类霍尔能量传输。
We study energy propagation along line-defects (edges) in two dimensional continuous, energy preserving periodic media. The unperturbed medium (bulk) is modeled by a honeycomb Schroedinger operator, which is periodic with respect to the triangular lattice, invariant under parity, P, and complex-conjugation, C. A honeycomb operator has Dirac points in its band structure: two dispersion surfaces touch conically at an energy level, E D [25],[27]. Periodic perturbations which break P or C open a gap in the essential spectrum about energy E D. Such operators model an insulator near energy E D. Our edge operator is a small perturbation of the bulk and models a transition (via a domain wall) between distinct periodic, P or C breaking perturbations. The edge operator permits energy transport along the line-defect. The associated energy channels are called edge states. They are time-harmonic solutions of the underlying wave equation, which are localized near and propagating along the line-defect. They are of great scientific interest due to their remarkable stability, and are a key property of topological insulators. We completely characterize the edge state spectrum within the bulk spectral gap about E D. At the center of our analysis is an expansion of the edge operator resolvent for energies near E D. The leading term features the resolvent of an effective Dirac operator. Edge state eigenvalues are poles of the resolvent, which bifurcate from the Dirac point. The corresponding eigenstates have the multiscale structure identified in [23]. We extend earlier work on zigzag-type edges [14] to all rational edges. We elucidate the role in edge state formation played by the type of symmetry-breaking and the orientation of the edge. We prove the resolvent expansion by a new direct and transparent strategy. Our results also provide a rigorous explanation of the numerical observations in [22],[38]; see also the photonic experimental study in [42]. Finally we discuss implications for the Valley Hall Effect, which concerns quantum Hall-like energy transport in honeycomb structures.