Topological and C*-Algebraic Quantum Matter

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，而且大系统也可以以量子方式运行。物理学家和数学家才刚刚开始了解物质的大量量子相：这些系统在微观上看起来不同，但在宏观上却有共同的特性。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，预计该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。该奖项还将通过培养研究生为美国的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建立一个框架来研究凝聚态物理和拓扑之间的关系。在凝聚态物理中，配备时间演化或哈密顿量的量子系统通常使用晶格模型来表示：这些是材料在低温下的简化模型。当量子系统的能隙非零时，某些特性对系统的变形具有鲁棒性，从而导致系统等效类的概念，即物质的间隙相。一些被称为不可逆的间隙相，被认为可以很好地近似于拓扑量子场论，尽管这种联系还没有得到严格的理解。然而，Kitaev提出了拓扑和凝聚态物理之间更基本的联系，认为可逆的间隙相可以用同伦理论意义上的环谱来分类。项目团队将通过引入可逆间隙相位的上同调理论来精确地做到这一点。在物理上，这对应于研究具有允许变化参数的量子系统，具有一些预定的限制，并且不改变系统的基本性质。该奖项支持旨在构建该理论的研究，并回答物质间隙相数学物理中的相关基本问题。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。

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.