Interacting topological phases and operator algebras

相互作用的拓扑相和算子代数

• 批准号: 19K14548
• 负责人: ボーン クリストファー・ジャック
• 金额: $ 2.33万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Early-Career Scientists
• 财政年份: 2019
• 资助国家: 日本
• 起止时间: 2019-04-01 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-19K14548/
• 关键词: Index theory; Quantum walk; Topological phases; Operator Algebras; Gapped ground state; SPT phase; operator algebras; gapped ground state; ground state; spectral flow; K-theory

The project has been quite successful in adapting techniques from operator algebras, index theory and noncommutative geometry to certain classes of quantum mechanical Hamiltonians and gapped ground states.Building from our established framework, we began a systematic study of topological and index theoretic properties of quantum walks, which can be regarded as quantum (discrete) analogues of random walks. In particular, quantum walks are expected to be useful in the implementation of algorithms in quantum computing. Determining robust properties of such quantum walks via a concrete link to topological invariants is therefore of significant value.We were able to adapt our mathematical techniques in the study of Hamiltonians and gapped ground states to quantum walk systems and define several topologically robust indices. This work has been submitted for publication and is currently under review.

该项目在将算子代数、指数理论和非对易几何的方法应用于某些量子力学哈密顿量和带隙基态方面取得了相当成功的成果。在我们建立的框架下，我们开始系统地研究量子行走的拓扑和指数理论性质，量子行走可以被看作是随机行走的量子（离散）模拟。特别是，量子行走预计将在量子计算中的算法的实现是有用的。通过一个具体的链接到拓扑不变量确定这样的量子行走的鲁棒性能，因此是显着的values.We能够适应我们的数学技术在研究的哈密顿量和带隙基态量子行走系统，并定义了几个拓扑鲁棒指数。这项工作已提交出版，目前正在审查中。

项目成果

The KO-valued spectral flow for skew-adjoint Fredholm operators

斜伴 Fredholm 算子的 KO 值谱流

• DOI: 10.1142/s1793525320500557
• 发表时间: 2020
• 期刊: Journal of Topology and Analysis
• 影响因子: 0.8
• 作者: Bourne Chris;Carey Alan L.;Lesch Matthias;Rennie Adam
• 通讯作者: Rennie Adam

Locally equivalent quasifree states and the coarse index

局部等效准自由态和粗指数

• 发表时间: 2021
• 作者: Bourne Chris;Schulz-Baldes Hermann;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne
• 通讯作者: Chris Bourne

Fermionic gapped ground states, topological phases and index theory

费米子有隙基态、拓扑相和指数理论

• 发表时间: 2022
• 作者: Bourne Chris;Schulz-Baldes Hermann;Chris Bourne;Chris Bourne;Chris Bourne
• 通讯作者: Chris Bourne

KK-theory and free fermionic topological phases

KK 理论和自由费米子拓扑相

• 发表时间: 2021
• 作者: Bourne Chris;Schulz-Baldes Hermann;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne
• 通讯作者: Chris Bourne

Spectral flow and topological phases

谱流和拓扑相

• 发表时间: 2020
• 作者: Bourne Chris;Schulz-Baldes Hermann;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne;Chris Bourne
• 通讯作者: Chris Bourne