Aperiodic Topological Materials and Meta-Materials

材料研究部和数学科学部为本奖项提供资金。它支持理论、计算和数学研究，目的是利用数学中的复杂概念，对材料中电子物质的拓扑状态及其发生有更深入、更精确的理解。在物质的热力学稳定的凝聚相中，原子自发地以决定材料的许多物理特性的模式排列。被研究最多的材料是晶体，其中原子以空间周期性模式排列，就像教室里的桌子一样。其他凝聚态相，如合金、非晶固体、准晶体和液体，显示出非周期原子结构。一般来说，周期性的丧失使得它们在理论上和实验上都更难以表征。然而，它们之所以令人感兴趣，是因为非周期结构比周期结构要多得多，这就提供了更多的方法来调整材料的物理性质，并使材料中的电子达到新的物质状态。除了自然发生的凝聚态相外，人们对合成材料在微观和纳米尺度上的图案越来越感兴趣。这些材料能够更可控地探索新特性、物质的新状态，以及由非周期性带来的相关现象。材料中的电子在原子核产生的能量景观中导航；它们受原子排列方式的强烈影响。电子也通过库仑很久以前发现的著名的静电相互作用相互作用。在许多情况下，静电相互作用的影响可以包含在原子势的变化中。当这是不可能的，电子被认为是强相关的，他们的动力学变得非常难以量化，即使是最强大的计算机。该奖项支持理论和计算研究，这些研究将探索电子在强非周期性和相关性条件下的动力学。重点将是检测那些在外部条件的微小修改下不会改变的动态特征，这些特征被称为拓扑特征。PI已经将算子代数、k理论和非交换几何等纯数学领域的思想转化为材料科学分析的先驱工具。它们在过去已经成功地应用于强无序拓扑环境中。在这个项目中，PI将进一步将这些工具扩展到一般的非周期和相关材料，以预测和系统地表征电子物质的新拓扑相和相关的物理效应。这些研究活动的最终目的是建立准确和有效的方法和概念来描述材料中极其复杂的电子系统。设想这些方法将提供所有可能的拓扑状态类别的全局视图和确保其稳定性的精确条件，并导致具体的计算机算法来定位这一全局图像中的任何合成图案材料。该奖项支持研究生培养成为应用非交换几何的未来专家，目前在理论凝聚态和材料物理中是一种罕见的能力。它还将支持PI编写的讲座、教学评论和教科书，以便在凝聚态物理学界的理论和实验成员中传播他的思想。技术摘要材料研究部和数学科学部为该奖项提供资金。它支持理论、计算和数学研究，目的是利用数学中的复杂概念，在大量非周期相关材料中系统化地搜索拓扑相和相关现象。算子代数，k理论和非交换几何将在这一追求中使用。一般目标是：对缺口体哈密顿量进行分类并探索其物理反应；建立k理论体边界对应原理，利用拓扑边界谱检测系统；并使用数值算法来实现非交换几何的思想，以举例说明和生成现实世界的概念和材料。人们早就知道，图案材料的体动力学发生在一个定义良好的物理可观察物代数中，它来源于图案的外壳。这是一个与给定模式相关联并由给定模式构建的拓扑空间。它增加了物理维度，可以极大地丰富体边界对应。例如，特殊图案的一维系统可以在二维中显示整数量子霍尔效应的物理特性，并显示拓扑量子泵浦。同样，特殊的二维系统可以在四维中表现整数量子霍尔效应的物理特性，并显示量子化的压电效应。这些系统在存在相关性的情况下，由于它们的降维，在理论上和数值上都更容易研究，因此对相关非周期系统的分类和彻底表征变得可行。当物理可观测代数的k理论可以明确计算时，拓扑相的分类变得详尽，k群的生成器提供了所有基本拓扑模型。非交换几何被发现提供了必要的工具来定义拓扑不变量，并将它们与物理响应联系起来，以及探索它们在谱间隙被迁移率间隙取代的制度中的命运。这个项目将把这些一般的想法整合到一个材料科学的搜索和发现项目中，这个项目远远超出了无序晶体的范畴，涵盖了物质的许多其他热力学相，算法合成的超材料和自然界中发现的复杂系统。该计划旨在提供准确和高效的数值算法，这些算法将系统地用于将理论进展引导到实验实验室。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。

NONTECHNICAL SUMMARYThe Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. It supports theoretical, computational, and mathematical research with an aim to use sophisticated concepts from mathematics to develop a deeper and more precise understanding of topological states of electronic matter in materials and their occurrence. In thermodynamically stable condensed phases of matter, atoms spontaneously arrange in patterns that determine many of the physical characteristics of the materials. The most studied materials are crystals, where atoms arrange themselves in spatially periodic patterns, like desks in a classroom. Other condensed matter phases such as alloys, amorphous solids, quasicrystals, and liquids display aperiodic atomic structures. In general, the loss of periodicity makes these more difficult to characterize, both theoretically and experimentally. Yet they are of interest because there are many more aperiodic than periodic structures, offering more ways to tailor the physical properties of the material and for the electrons in the material to achieve new states of matter. Besides the naturally occurring condensed matter phases, there is a growing interest in synthetic materials patterned on the microscopic and nanoscopic length scales. These materials enable a more controlled exploration of new properties, new states of matter, and related phenomena enabled by aperiodicity.The electrons in materials navigate the energy landscape generated by the atomic cores; they are strongly influenced by the pattern in which the atoms are arranged. The electrons also interact with each other through the celebrated electrostatic interaction discovered by Coulomb long ago. In many instances, the effect of the electrostatic interaction can be subsumed in modified atomic potentials. When this is not possible, the electrons are said to be strongly correlated and their dynamics become very difficult to quantify, even with the most powerful computers.This award supports theoretical and computational research which will explore the dynamics of the electron in conditions of both strong aperiodicity and correlation. The focus will be on detecting those dynamical characteristics which do not change under small modifications of the external conditions and which are said to be topological. The PI has adapted ideas from fields of pure mathematics, such as Operator Algebras, K-theory and Non-Commutative Geometry, into pioneering tools of analysis in materials science. They have been successfully used in the past in the context of strongly disordered topology. In this project, the PI will further extend these tools to generic aperiodic and correlated materials to predict and systematically characterize novel topological phases of electronic matter and related physical effects.These research activities are ultimately aimed to build accurate and efficient methods and concepts to describe extremely complex electronic systems in materials. It is envisioned that the methods will provide a global view of all possible classes of topological states and the precise conditions that ensure their stability and lead to concrete computer algorithms to locate any synthetically patterned material in this global picture. The award supports graduate student training to become the future experts in applied Non-Commutative Geometry, currently a rare ability in theoretical condensed matter and materials physics. It will also support lectures, pedagogical reviews and textbooks prepared by the PI as an effort to disseminate his ideas among theoretical as well as experimental members of our condensed matter physics community.TECHNICAL SUMMARYThe Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. It supports theoretical, computational, and mathematical research with an aim to use sophisticated concepts from mathematics to systematize the search for topological phases and related phenomena in the vast classes of aperiodic correlated materials. Operator Algebras, K-theory and Non-Commutative Geometry will be utilized in this pursuit. The generic goals are: to classify the gapped bulk Hamiltonians and explore their physical responses; establish K-theoretic bulk-boundary correspondence principles and detect those systems with topological boundary spectra; and to use numerical algorithms that implement ideas from Non-Commutative Geometry to exemplify and generate real-world concepts and materials. It has been long known that the bulk dynamics of a patterned material occur inside a well-defined algebra of physical observables, which derives from the hull of the pattern. This is a topological space associated with and constructed from the given pattern. It augments physical dimensions and can greatly enrich the bulk-boundary correspondence. For example, specially patterned 1-dimensional systems can manifest the physics of the Integer Quantum Hall Effect in 2-dimensions and display topological quantum pumping. Similarly, specially patterned 2-dimensional systems can manifest the physics of the Integer Quantum Hall Effect in 4-dimensions and display the quantized piezo-magneto-electric effect. These systems are much easier to investigate, both theoretically and numerically, in the presence of correlations because of their reduced dimensionality, hence the classification and thorough characterization of correlated aperiodic systems becomes feasible. When the K-theories of the algebras of physical observables can be computed explicitly, the classification of topological phases becomes exhaustive, with the generators of the K-groups providing all fundamental topological models. Non-Commutative Geometry was found to provide the necessary tools to define topological invariants and to connect them to physical responses, as well as to explore their fate in regimes where spectral gaps are replaced by mobility gaps. This project will integrate these general ideas into a search-and-discovery program in materials science that goes well beyond the class of disordered crystals and covering many other thermodynamic phases of matter, algorithmically synthesized meta-materials and complex systems found in nature. This program is aimed to provide accurate and efficient numerical algorithms, which will be systematically used to channel theoretical advances to experimental laboratories.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.