Topology and Topological Phases in Strongly Interacting Many-Body Systems

技术摘要材料研究部和数学科学部为该奖项提供资金。它为强相互作用凝聚态多体系统的理论研究和教育提供了支持。主要目的是深入了解拓扑相的起源和行为。这些相是物质的状态，它们不是自发对称性破缺的产物，因此不具有序参量，而是具有序参量。量子点菜了吗?其中基态简并度是由它们所在空间的拓扑结构决定的。这种拓扑相位在量子计算中有潜在的应用，量子时间演化中固有的大量并行性为传统机器上需要指数时间的问题提供了快速解决方案。PI将进一步探索Calogero-Sutherland模型与量子霍尔流体之间的对应关系，旨在理解非阿贝尔量子流体中边缘态和体态之间的关系。PI将探索如何通过实验直接观察阿贝尔量子霍尔效应中的阴离子统计的新思路。研究还将扩展到石墨烯中曲率和电子结构的相互作用。该研究的具体目标有三个方面：了解多体波函数的局部特性如何共同产生表征这些系统的全局基态简并，探索这些系统在量子计算设备构建中的可能应用，并研究可能通过设计或适当的工程具有所需拓扑特性的新系统。多体系统的量子场论将与无限维电流代数的表示理论以及在适当情况下保证具有理想性质的微分方程解的存在性的指标定理一起进行研究。这个项目有助于建立量子计算机的长期目标，但也提供了研究生的培训。这些学生将通过获得有价值的数学和计算技能以及物理洞察力来整合研究和教育，这将为他们在学术界，工业界和国家实验室的研究事业做好准备。材料研究部和数学科学部为该奖项提供资金。它为凝聚态物理和数学的理论研究和教育提供支持。PI将使用先进的理论方法，以及数学和物理概念来探索量子力学系统中出现的一种新型自发组织的概念。众所周知的是有序状态的概念，就像原子在规则排列中自发组织形成完美的晶体材料一样。有序晶体态的对称性低于晶体生长的熔体。在量子力学系统中，一种新的秩序是可能的，在这种秩序中，对称性在进入有组织状态时不会发生变化。相反，这种组织更抽象地反映在量子力学波函数的拓扑特性中。这个概念是在研究垂直磁场中被限制在一个平面上的电子气体时变得明显的。PI将使用先进的理论技术和数学概念来进一步研究这一想法，并确定是否出现新的物质拓扑量子态，并可以在实验中直接观察到。令人兴奋的是，这种物质状态可以被操纵来执行本质上并行的计算操作，并且可以以比现有和可预见的超级计算机更高的速度执行某些算法。本研究为量子计算的智力基础及其实验实现做出了贡献。该奖项还支持教育。学生将通过获得有价值的数学和计算技能以及物理洞察力来整合研究和教育，这将为他们加入21世纪的劳动力做好准备。

TECHNICAL SUMMARYThe Division of Materials Research and the Mathematical Sciences Division contribute funds to this award. It provides support for theoretical research and education on strongly interacting condensed matter many-body systems. The principal objective is to gain insight into the origin and behavior of topological phases. These phases are states of matter that are not the product of spontaneous symmetry breaking, and therefore do not have an order-parameter, but possess instead ?quantum order? in which the ground state degeneracy is determined by the topology of the space in which they live. Such topological phases have potential applications in quantum computing where the massive parallelism inherent in quantum time evolution provides fast solutions for problems that would require exponential time on conventional machines.The PI will further explore the correspondence between Calogero-Sutherland models and quantum Hall fluids with an aim to understanding the relation between edge and bulk states in non-abelian quantum fluids. The PI will explore new ideas for how to directly observe anionic statistics in the abelian quantum Hall effect through experiments. Research will also extend to the interplay of curvature and electronic structure in graphene. The specific aims of the research are three-fold: to understand how the local properties of the many-body wave-functions conspire to produce the global ground-state degeneracy that characterizes these systems, to explore possible applications of these systems for the construction of quantum computation devices, and to investigate novel systems that may, by design or by suitable engineering, possess the desired topological properties.Quantum field theory of many-body systems will be used to carry the research along with the representation theory of infinite dimensional current algebras, and index theorems that, under suitable circumstances, guarantee the existence of solutions of differential equations with desirable properties.This project contributes to the long-term goal of building a quantum computer, but also provides training of graduate students. These students will integrate research and education by acquiring valuable mathematical and computational skills, along with physical insight, that will prepare them for research careers in academia, industry, and the national laboratories.NON-TECHNICAL SUMMARYThe Division of Materials Research and the Mathematical Sciences Division contribute funds to this award. It provides support for theoretical research and education at the interface of condensed matter physics and mathematics. The PI will use advanced theoretical methods, and mathematical and physical concepts to explore the notion of a new kind of spontaneous organization that appears in quantum mechanical systems. Well known is the notion of ordered states like the spontaneous organization of atoms in a regular array to form perfect crystalline materials. The ordered crystal state has a lower symmetry than the melt from which the crystal grows. In quantum mechanical systems, a new kind of order is possible in which there is no change in symmetry upon going to the organized state. Rather, the organization is reflected more abstractly in the topological properties of the quantum mechanical wavefunction. This notion became apparent from the study of a gas of electrons confined to a plane in a perpendicular magnetic field. The PI will use advanced theoretical techniques and mathematical concepts to further study this idea and to determine if a new topological quantum state of matter appears and can be directly observed in experiments. The exciting possibility is that this state of matter can be manipulated to perform computational operations that are intrinsically parallel and can execute certain algorithms at much higher speed than existing and foreseeable supercomputers. This research contributes to the intellectual foundations of quantum computing and toward its experimental realization. This award also supports education. Students will integrate research and education by acquiring valuable mathematical and computational skills, along with physical insight, that will prepare them to join the workforce of the 21st century.