From Higher-Order Topological Superconductors to Topological Quantum Codes.

在过去的十年里，一种被称为拓扑绝缘体的新材料的发现引起了极大的兴奋，因为它有可能实现新的现象，例如具有高导电性表面的绝缘材料。这些令人惊讶的特性的关键在于这些系统的拓扑结构，即整体的、量化的、对局部缺陷(如缺陷和杂质)不敏感的材料特性。在描述这类材料方面的理论进步允许对另一类多体系统进行拓扑分类，即拓扑超导体(TSCs)。虽然TSCs的分类现在已经相当成熟，但目前的注意力已经转移到TSCs的一个新的子类，即高阶TSCs。这些系统表现出更多源自空间对称性的异国情调。在某些情况下，人们可以预言存在导致非局域支持的基态简并的奇异角界零能天体的存在。新的例子表明，对于某些形式的相互作用，这种系统还可能支持与拓扑量子码密切相关的拓扑有序相。这种编码构成了最有前途的量子纠错方案之一，并可能导致量子计算机对最相关的扰动源具有较强的鲁棒性。本项目的目标是发展一种高阶TSC的分类，并在此基础上研究相互作用的高阶TSC与拓扑量子码之间的联系。我们将在现有理论框架的基础上，进一步发展，以表征拓扑材料和相关形式的拓扑秩序。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.