Entanglement and emergence in new quantum states of matter

非技术总结该奖项支持理论研究和教育进入物质的新阶段，称为拓扑阶段。正如朗道发现的，物质的状态，如液体和晶体，可以通过了解在给定的状态下可以执行什么对称变换来组织，从而使其保持不变。例如，围绕食盐晶体的主轴旋转90度，会将新原子旋转到原来由同一种原子占据的位置，因此晶体看起来没有变化。对称性的概念也允许通过类似的考虑来组织磁状态和其他状态。近年来，人们发现了拓扑学的思想，拓扑学是数学的一个分支，涉及通过变形、扭曲和拉伸对象而不变的几何性质，它带来了对物质的新的可能的阶段的洞察，称为拓扑阶段。其中包括拓扑绝缘体，它们与普通绝缘体的根本不同之处在于，虽然主体不导电，但它们的表面导电，就像它们属于金属一样。该奖项支持理论研究，将利用金属表面态等拓扑现象来发展拓扑相的系统理论。量子纠缠有望在这一理论中发挥关键作用，这将使寻找、发现和设计新的拓扑态成为可能。量子纠缠指的是这样一种现象，即系统中各个组件的状态，即使这些组件在空间上可能相距很远，也不能相互独立地描述；作为量子力学的结果，各个组件的行为是相互关联的。拓扑量子纠缠导致了可能的惊人现象，包括电子的一小部分电荷，处于电子和光子之间的基本意义上的粒子-光的量子，以及对光和电子起源的更统一的理解。可以发现或设计新的拓扑态，这些拓扑态可以形成利用操纵量子力学状态的计算的基础，即拓扑量子计算机。该奖项支持理论研究和教育，目的是发现新的理论概念和方法，以更精确地发展拓扑相和对称保护拓扑(SPT)相的综合理论。这项研究包括四个部分：(1)开发一个系统来标记拓扑/SPT阶段；(2)发现一组能够测量和区分不同拓扑阶段的物理测量；(3)创建一个允许对所有拓扑阶段进行分类的数学理论；(4)开发有效的方法来计算包含拓扑/SPT阶段的一般系统的相图。这项研究可能导致发现一类新的量子材料--高度纠缠的拓扑材料，这可能为利用量子纠缠实现一类新的器件概念和电子和光学器件提供基础。该理论的发展可能需要或刺激新数学的创造，并可能导致对从量子计算到基本粒子等领域的基本见解。

NONTECHNICAL SUMMARYThis award supports theoretical research and education into new phases of matter, called topological phases. As Landau discovered, states of matter such as liquids and crystals can be organized through understanding what symmetry transformations can be performed on a given state that leaves it unchanged. For example, a rotation of 90 degrees around a principle axis of a crystal of common salt, rotates new atoms into positions that were originally occupied by the very same kind of atom, so the crystal appears unchanged. The concept of symmetry also allows magnetic and other states to be organized by similar considerations. In recent years, it has been discovered that ideas from topology, the branch of mathematics concerned with geometric properties that are unchanged by deformations, twisting, and stretching objects, brings insight into new possible phases of matter, called topological phases. Among these are topological insulators that are fundamentally different from ordinary insulators in that while the bulk does not conduct electricity, their surfaces do, as if they belonged to a metal.This award supports theoretical research that will use topological phenomena like the metallic surface states to develop a systematic theory of topological phases. Quantum entanglement is expected to play a crucial role in this theory which will enable the search, discovery, and design of new topological states. Quantum entanglement refers to the phenomenon whereby states of individual components of a system, eventhough the components may be spatially separated by large distances, cannot be described independently of each other; as a consequence of quantum mechanics, the individual components act in a correlated way.Topological quantum entanglement leads to possible amazing phenomena, including electric charge that is a fraction of an electron charge, particles that are in a fundamental sense between electrons and photons - the quantum of light, and a more unified understanding of the origin of light and electrons. New topological states may be discovered or designed that can form the foundations for computation that utilizes the manipulation of quantum mechanical states, a topological quantum computer.TECHNICAL SUMMARY This award supports theoretical research and education with the aim of discovering new theoretical concepts and methods to more precisely develop a comprehensive theory of topological phases and symmetry protected topological (SPT) phases. The research has 4 parts: (1) develop a system to label topological /SPT phases; (2) discover a set of physical measurements that can measure and distinguish different topological phases; (3) create a mathematical theory that allows for the classification of all topological phases; (4) develop effective methods to enable the calculation of the phase diagram of a generic system that contains topological/SPT phases. This research may lead to the discovery of a new class of quantum materials, highly entangled topological materials, which may provide the foundations for a new class of device concepts and electronic and optical devices made possible by exploiting quantum entanglement. The development of the theory may require or stimulate the creation of new mathematics and may lead to fundamental insights into areas extending from quantum computing to elementary particles.