非可換幾何学による物質のトポロジー相の解明

使用非交换几何阐明材料的拓扑相

• 批准号: 16F16728
• 负责人: 小谷 元子
• 金额: $ 1.28万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2016
• 资助国家: 日本
• 起止时间: 2016-11-07 至 2019-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16F16728/
• 关键词: Topological phases; operator algebras; Kasparov theory; aperiodic media; Topological phases; operator algebras; Kasparov theory; aperiodic media

We have previously identified Delone sets as a useful model for atomic configurations at low temperature and without any assumptions on the periodicity or structure of the material. We have now developed a framework to understand the topological phases of systems in material science and meta-materials that can be modelled by a Delone configuration. In recent work, we were able to prove the quantization of the so-called bulk complex topological phases of Delone materials. This result gives a prediction of conductivity properties of Delone and aperiodic lattices that appears to be new and novel. In particular, we provide a mathematical theorem and strong numerical evidence that a quantum Hall like effect is possible in aperiodic and amorphous metals, e.g. metallic glass. Interesting future research would be to investigate whether such properties can be experimentally realized. This paper was submitted in December 2017 and is currently under peer review.We are also finalizing work that more comprehensively characterizes the topological properties of Delone lattices and topological phases using K-theory. This includes topological phases with anti-linear symmetries such as time reversal symmetry. We also prove the bulk-boundary correspondence and study edge properties of Delone topological materials. We expect to complete this work within the coming weeks.

我们先前已经将Delone集识别为在低温下原子构型的有用模型，并且没有对材料的周期性或结构的任何假设。现在，我们已经开发了一个框架，以了解可以通过DeLone配置建模的材料科学和元材料中系统的拓扑阶段。在最近的工作中，我们能够证明Delone材料所谓的散装复杂拓扑阶段的量化。该结果可以预测DeLone和Aperiodic Lattices的电导率特性，这些特性似乎是新颖而新颖的。特别是，我们提供了数学定理和强有力的数值证据，即在大部分和无定形金属中可以使用量子大厅的效应，例如金属玻璃。有趣的未来研究是研究是否可以实验实现此类特性。本文于2017年12月提交，目前正在同行评审中。我们还在完成更全面地描述使用K理论的Delone Lattices和拓扑阶段的拓扑特性。这包括具有反线性对称性的拓扑阶段，例如时间逆转对称性。我们还证明了Delone拓扑材料的庞大的对应关系和研究边缘特性。我们希望在未来几周内完成这项工作。