Waves and Topology in Quantum Materials

设计具有革命性性能的材料是工业和应用研究的中心主题。原型包括拓扑绝缘体（非常坚固的单向通信设备，有望稳定量子计算）、隐形斗篷（使光线远离自身的纺织品）和原子薄电池（如吸收并再利用Wi-Fi或电信波能量的涂料）。该项目旨在(a)开发定量工具来预测量子粒子在这种定制环境中的行为；(b)将这些工具应用于具有最佳异常效果的工程材料；(c)培养具有扎实量子科学和定量分析背景的工作人员。这些目标可以通过增强对量子现象的理解来帮助设计新一代技术设备。重点将放在三个在技术领域具有很高前景的主题上：(i)沿着拓扑绝缘体边缘的能量不对称传播；（二）伪磁性的产生和影响；（iii）薄层材料中的态密度。第一个主题定量地分析了拓扑结构不同的材料之间能量传输的特征：波的群速度，不对称通道的轮廓，扰动的影响。第二个主题从数学上研究定制材料中的朗道能级和量子霍尔效应，这些材料可以产生经典实验无法达到的强度的人工磁场。最后一个主题是扭曲双层石墨烯，这是一种革命性的结构，对于特定的调谐参数，它表现出高密度的状态包装和超导性。这些现象的数学研究依赖于发展和应用各种复杂的分析工具，如周期算符的半经典微积分，狄拉克类系统中的散射理论和畸变平面波，准周期结构的多尺度和均匀化技术，非自伴随算符的谱理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Designing materials with revolutionary properties is a central theme of industrial and applied research. Prototypes include topological insulators (extraordinarily robust one-way communication devices that promise to stabilize quantum computations), invisibility cloaks (textiles that deviate light away from themselves) and atomic-thin batteries (such as paints that absorb and reutilize energy from Wi-Fi or telecommunication waves). This project aims to (a) develop quantitative tools to predict the behavior of quantum particles within such tailored environments; (b) apply these tools to engineer materials with optimized anomalous effects; and (c) train a workforce with a solid background in both quantum science and quantitative analysis. These objectives can aid the design of a new generation of technological devices via an enhanced understanding of quantum phenomena.The focus will be on three themes that carry high promises in technological fields: (i) the asymmetric propagation of energy along edges of topological insulators; (ii) the generation and the effects of pseudo-magnetism; and (iii) the density of states in thin-layer materials. The first topic quantitatively analyzes the characteristics of energy transport between topologically distinct materials: the group velocity of waves, the profile of asymmetric channels, the influence of perturbations. The second topic investigates mathematically Landau levels and quantum Hall effects in custom-made materials that enact artificial magnetic fields of intensities inaccessible to classical experiments. The last topic focuses on twisted bilayer graphene, a revolutionary structure that - for specifically tuned parameters - exhibits high-density state-packing and superconductivity. The mathematical investigation of these phenomena relies on developing and applying diverse and sophisticated analytic tools such as semiclassical calculus for periodic operators, scattering theory and distorted plane waves in Dirac-like systems, multiscale and homeogenization techniques of quasiperiodic structures, and spectral theory of non-selfadjoint operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.