From Local to Global Properties in Strongly Interacting Many-Body Systems

技术概述：材料研究部和数学科学部在nsf数学科学优先领域下为该奖项提供资金，并通过为未来网络基础设施提供基础的基础研究为网络基础设施做出贡献。该奖项支持强相互作用凝聚态多体系统理论的研究和教育。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，不是自发对称破缺的产物，因此没有秩序参数，而是具有“量子秩序”，其中基态简并是由它们所处空间的拓扑结构决定的。该项目解决了当前基础物理领域的一系列重要和困难的问题，这些问题非常活跃。它将导致物理学和数学的进步。提出的工作的主要目标有两个方面：首先，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，其次，探索解决量子计算问题的最后一部分的可能策略，即将孤立的量子CPU重新连接到经典世界并读取计算的答案。该方法将使用多体系统的量子场论与无限维李代数的表示理论相结合。更广泛的影响：在这个项目中研究的拓扑阶段可能会应用于量子计算。即使是简单量子系统的数值模拟也需要指数级大的存储空间和指数级长的计算时间。N个自旋1/2粒子（或量子位）集合的量子态需要2N个经典变量来描述，求解薛定谔方程以遵循其时间演化需要一台传统计算机每个时间步执行多达2N个操作。由量子组件构建的计算机将能够利用量子时间演化中固有的大量并行性，为传统机器需要指数级时间的问题提供快速解决方案。建造量子计算机需要克服的困难有很多。其核心的量子系统必须与环境强隔离，以避免退相干；构成基本计算步骤的酉变换必须以足够的精度执行，以使纠错码保持有效；而且，在完成所有量子计算之后，系统必须能够以一种不干扰结果的测量过程读取输出的方式重新连接到外部世界。拓扑相基态的内部希尔伯特空间具有量子计算机CPU所需的许多特征，除了易于读出之外。这个项目解决了最后一个问题，也是内部希尔伯特空间本身的起源。非技术概述：材料研究部和数学科学部在nsf数学科学优先领域下为该奖项提供资金，并通过为未来网络基础设施提供基础的基础研究为网络基础设施做出贡献。该奖项支持强相互作用凝聚态多体系统理论的研究和教育。这项研究涉及一个重要的基本问题：物质的状态是如何组织起来的？这个问题的部分动机是对量子霍尔系统和自旋液体的研究，这些研究揭示了物质的状态不能简单地由对称和其他原理组织起来，这些原理是描述物质状态之间转换的强大标准理论的基础。这些拓扑相，正如他们所说的那样，可能是凝聚态物理前沿的一些最具挑战性的问题的基础，例如在高温超导体和相关材料中观察到的不寻常状态的性质，以及如何组织生活在各种复杂量子力学系统边界的边缘状态并在低温下控制其性质。PI将把理论凝聚态物理的先进量子力学概念和尚未达到理论凝聚态物理主流的复杂数学方法集中在模型系统上，以深入了解这些不寻常的物质状态及其组织方式。拓扑相的操作是实现基于量子力学的强大的新计算形式的稳健实现的有希望的途径。PI还将参与实现量子计算的障碍，即如何在宏观世界中访问由量子力学设备操作产生的信息。该研究对数学领域和理论高能物理都有潜在的影响。

TECHNICAL SUMMARY:The Division of Materials Research and the Division of Mathematical Sciences contribute funding to this award under the NSF-wide Mathematical Sciences Priority Area and contributes to cyberinfrastructure through fundamental research that provides the foundations of future cyberinfrastructure. This award supports research and education on the theory of strongly interacting condensed matter many-body systems. The 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.The project addresses a range of important and difficult problems in currently fertile and therefore very active, areas of fundamental physics. It should lead to progress both in physics and in mathematics. The principal goals of the proposed work are two-fold: firstly, to understand how the local properties of the many-body wave-functions conspire to produce the global ground-state degeneracy that characterizes these systems, and, secondly, to explore a possible strategy for addressing the last part of the quantum computing problem that of reconnecting the isolated quantum CPU to the classical world and reading of the answer to the computation.The method will be to use the quantum field theory of many-body systems in combination with the representation theory of infinite dimensional Lie algebras.Broader Impact: The topological phases being studied in this project may have applications to quantum computing. The numerical simulation of even simple quantum systems requires exponentially large storage space and exponentially long computation times. The quantum state of a collection of N spin- 1/2 particles, or qubits, requires 2N classical variables for its description, and solving the Schrdinger equation to follow its time evolution requires a conventional computer to perform up to 2N operations per time step. A computer built out of quantum components would be able to exploit the massive parallelism inherent in quantum time evolution to provide fast solutions for problems that would require exponential time on conventional machines. The difficulties to be overcome in building a quantum computer are many. The quantum system at its core must be strongly isolated from the environment so as to avoid decoherence; the unitary transformations that constitute the elementary computational steps must be performed with sufficient precision that error-correcting codes remain effective; and, after all the quantum computation has been performed, the system must be capable of being reconnected to the outside world in such a way that the output can be read off via a measurement process that does not disturb the result. The internal Hilbert space of the topological-phase ground states has many of the features desired of a quantum computer CPU, with the exception of ease of read-out. This project addresses this last problem, and also the origin of the internal Hilbert space itself.NON-TECHNICAL SUMMARY:The Division of Materials Research and the Division of Mathematical Sciences contribute funding to this award under the NSF-wide Mathematical Sciences Priority Area and contributes to cyberinfrastructure through fundamental research that provides the foundations of future cyberinfrastructure. This award supports research and education on the theory of strongly interacting condensed matter many-body systems. The research engages an important fundamental question: How are states of matter organized? This question is motivated in part by studies of quantum Hall systems and spin liquids that reveal states of matter that cannot be simply organized by symmetry and other principles that underlie the powerful standard theory that describes transformations among states of matter. These topological phases, as they are called, may underlie some of the most challenging problems at the frontiers of condensed matter physics, such as the nature of unusual states observed in high temperature superconductors and related materials, and how edge states that live at the boundaries of various complex quantum mechanical systems and control their properties at low temperature are organized. The PI will focus advanced quantum mechanical concepts of theoretical condensed matter physics and sophisticated mathematical methods that have not yet reached the mainstream of theoretical condensed matter physics on model systems to gain insight into these unusual states of matter and how they are organized. The manipulation of topological phases is a promising route to the realization of a robust implementation of a powerful new form of computation based on quantum mechanics. The PI will also engage a barrier to the realization of quantum computation, how to access in the macroscopic world information that is created by the operation of a quantum mechanical device. The research has potential impact on the field of mathematics and also on theoretical high energy physics.