Topological and C*-Algebraic Quantum Matter

拓扑和 C*-代数量子物质

• 批准号: 2055501
• 负责人: Markus Pflaum
• 金额: $ 38.59万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055501&HistoricalAwards=false
• 关键词: Topological; Algebraic; Quantum; Matter

A series of revolutionary developments across physics reveal that the phenomena of quantum mechanics not only play a crucial role for very small systems, but that large systems can also behave in a quantum way. Physicists and mathematicians are only beginning to understand a vast array of quantum phases of matter: classes of systems that may look different microscopically but share common macroscopic properties. This project aims to solidify the connection between topological phases of matter as thought of by condensed matter physicists and their conjectured mathematical classification. The intended approach to this problem is expected to have immediate applications to condensed matter physics and quantum information science via new examples of topological phases and the development of new theoretical methods. The award will also contribute to US workforce development through the training of graduate students. The principal scientific goal of this project is to use algebraic quantum mechanics to develop a framework to study the relationship between condensed matter physics and topology. In condensed matter physics quantum systems equipped with a time evolution or Hamiltonian are often presented using lattice models: these are simplified models of a material at low temperature. When the energy gap of a quantum system is non-zero, some properties are robust to deformations of the system, leading to the notion of equivalence classes of systems known as gapped phases of matter. Some gapped phases, called invertible, are believed to be well approximated by topological quantum field theories, although this connection is not understood rigorously. However, Kitaev has proposed a more fundamental connection between topology and condensed matter physics, suggesting that invertible gapped phases are classified by a loop-spectrum in the sense of homotopy theory. The project team will make this precise by introducing a cohomology theory of invertible gapped phases. Physically, this corresponds to studying quantum systems with parameters that are allowed to vary, with some predetermined restrictions and without changing the fundamental properties of the system. This award supports investigations aimed toward constructing this theory and answering related fundamental questions in the mathematical physics of gapped phases of matter.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.

物理学领域的一系列革命性发展表明，量子力学现象不仅对非常小的系统起着至关重要的作用，而且大型系统也可以以量子方式表现。物理学家和数学家才刚刚开始理解物质的大量量子相：一类系统在微观上看起来不同，但具有共同的宏观性质。该项目旨在巩固凝聚态物理学家所认为的物质拓扑相与他们推测的数学分类之间的联系。通过拓扑相的新例子和新理论方法的发展，预期这个问题的预期方法将立即应用于凝聚态物理和量子信息科学。该奖项还将通过培训研究生为美国劳动力发展做出贡献。该项目的主要科学目标是利用代数量子力学开发一个框架来研究凝聚态物理与拓扑学之间的关系。在凝聚态物理学中，具有时间演化或哈密顿量的量子系统通常使用晶格模型：这些是低温下材料的简化模型。当量子系统的能隙非零时，某些性质对系统的变形具有鲁棒性，导致被称为物质的带隙相的系统的等价类的概念。一些被称为可逆的带隙相被认为是拓扑量子场论很好的近似，尽管这种联系并没有得到严格的理解。然而，Kitaev提出了拓扑学和凝聚态物理学之间的一个更基本的联系，认为可逆的有间隙的相位可以通过同伦理论意义上的环谱来分类。项目团队将通过引入可逆空位相位的上同调理论来精确地实现这一点。在物理上，这相当于研究允许参数变化的量子系统，具有一些预定的限制，并且不改变系统的基本属性。该奖项支持旨在构建该理论并回答物质间隙相的数学物理学中相关基本问题的研究。该奖项反映了NSF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而被认为值得支持。

项目成果

Continuous dependence on the initial data in the Kadison transitivity theorem and GNS construction

Kadison 传递性定理和 GNS 构造中对初始数据的持续依赖

• DOI: 10.1142/s0129055x22500313
• 发表时间: 2022
• 期刊: Reviews in Mathematical Physics
• 影响因子: 1.8
• 作者: Spiegel, Daniel;Moreno, Juan;Qi, Marvin;Hermele, Michael;Beaudry, Agnès;Pflaum, Markus J.
• 通讯作者: Pflaum, Markus J.