Interacting topological phases and operator algebras

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。

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.