From Higher-Order Topological Superconductors to Topological Quantum Codes.

在过去的十年里，一类被称为拓扑绝缘体的新材料的发现引起了极大的兴奋，因为它有可能实现诸如具有高导电性表面的绝缘材料等新现象。这些令人惊讶的特性的关键在于这些系统的拓扑结构，即块状材料对缺陷和杂质等局部缺陷不敏感的全局、量子化特征。描述这类材料的理论进步使得对另一类多体系统的拓扑分类成为可能，即拓扑超导体（TSCs）。虽然tsc的分类现在已经相当确定，但目前的许多注意力已经转移到tsc的一个新的亚类，即高阶tsc。这些系统显示出更多来自空间对称性的奇异特性。在某些情况下，人们可以预测存在奇异的角界零能量物体，导致非局部支持的基态简并。新的例子表明，对于某些形式的相互作用，这样的系统也可能支持与拓扑量子码密切相关的拓扑有序相。这种编码构成了量子纠错最有希望的方案之一，并可能导致量子计算机对大多数相关干扰源具有鲁棒性。本项目的目标是建立高阶TSCs的分类，并在此基础上研究相互作用的高阶TSCs与拓扑量子码之间的联系。我们将建立并进一步发展现有的表征拓扑材料和拓扑秩序相关形式的理论框架。我们计划用数值模拟来支持我们的正式理论发现。

The discovery of a new class of materials known as topological insulators has been cause for great excitement over the past decade owing to the possibility of realising novel phenomena such as insulating materials with highly conducting surfaces. The key to these surprising properties lies in the topology of these systems, that is, global, quantised, characteristics of the bulk material that are insensitive to local imperfections such as defects and impurities. The theoretical advancements in describing such materials have allowed for the topological classification of a further class of many-body systems, namely topological superconductors (TSCs).While the classification of TSCs is now fairly established, much current attention has shifted to a new subclass of TSCs known as higher-order TSCs. These systems display even more exotic properties deriving from spatial symmetries. Under certain circumstances, one may predict the existence of exotic corner-bound zero-energy objects that lead to a nonlocally supported ground state degeneracy. Novel examples indicate that, for certain forms of interactions, such systems may also support topologically ordered phases closely related to topological quantum codes. Such codes form one of the most promising schemes for quantum error correction and may lead to quantum computers robust against most relevant sources of disturbances.The goal of this project is to develop a classification of higher-order TSCs and, based on this, to investigate the connection between interacting higher-order TSCs and topological quantum codes. We will build on, and develop further, existing theoretical frameworks for characterising topological materials and correlated forms of topological order. We plan to support our formal theoretical findings with numerical simulations.