Interacting topological phases and operator algebras

该项目成功地将算符代数、指数理论和非对易几何的技术应用于某些量子力学哈密顿量和有隙基态。从我们建立的框架出发，我们开始系统地研究量子游动的拓扑和指数理论性质，量子游动可以被视为随机游动的量子(离散)类似物。特别值得一提的是，量子行走有望在量子计算算法的实现中发挥作用。因此，通过与拓扑不变量的具体联系来确定这种量子游动的健壮性具有重要的价值。我们能够将我们在哈密顿量和带隙基态研究中的数学方法应用到量子游动系统中，并定义了几个拓扑健壮性指数。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.