三维拓扑序的新奇物理性质研究

Topological order (TD) is characteristic of non-trivial quantum state, which can be used to realize quantum computer. Once discovered in the frequent quantum Hall state, it aroused extensive research. During about two decades, the 2D topological order, especially 2D Abelian topological order, have been deeply studied, and experimentalists have designed a lot of experiments to realize topological order. However, the theory of 3D topological order and non-Abelian topological order have not be totally completed. In this program, we will make effort in the research of 3D topological order and non-Abelian topological order. Beginning with several spin model such as 3D toric-code model and Kitaev model, we will firstly study the topological quantum phase transition of 3D topological order, the edge states and topological entanglement entropy to recognize it comprehensively. Then the quantum tunneling process of 3D topologica order will be researched . Using the high-order degenerate perturbation approach, the effective pseudo spin model will be derived, and the quantum manipulating of 3D topological order could be designed, which is important to the realistic of topological quantum computer. In the end, we will study the non-Abelian topological order. The quantum phase transition from Z2 topological order to non- Abelian phase , the edge states and quantum manipulating of non-Abelian topological order will be studied.

由于拓扑序的新奇性质和实现量子容错计算的应用前景，一经发现后就引起了人们的广泛研究。近二十年的研究探索，人们对二维拓扑序，尤其是二维阿贝尔拓扑序有了深入认识，实验学家们也设计了一系列实验来实现拓扑序。但是对三维拓扑序和非阿贝尔拓扑序还未建立起完善的研究。本项目将主要致力于三维拓扑序以及非阿贝尔拓扑序的研究。我们将从几个自旋模型出发，如三维toric-code 模型以及Kitaev 模型，首先研究三维拓扑序的拓扑量子相变、边界态、拓扑纠缠熵，全面认识三维拓扑序。其次，我们将研究三维拓扑序的量子隧道效应，利用我们发展的简并微扰方法，得到三维拓扑序的有效模型，实现对三维拓扑序的量子调控，这对实现拓扑量子计算具有重要意义。最后我们将主要研究非阿贝尔拓扑序，主要研究Z2拓扑序态到非阿贝尔拓扑序态的量子相变，以及非阿贝尔拓扑序的边界态、量子调控。

申请人在拓扑量子态的研究中取得了一定成果，在国际刊物上发表论文2篇。在项目执行中，我们主要研究了二维拓扑序问题，讨论了非厄米 Z2 拓扑序模型，对拓扑序边界态做了较为详尽的研究。我们提出了一类超冷费米原子s波拓扑超流，我们发现在此二维系统中，线位错的两端存在马约拉纳费米子，并且在超流涡旋中心也得到了零模。此外，我们还研究了二维拓扑超导体中的马约拉纳费米子问题。通过数值计算马约拉纳费米子体系的一级拓扑量子相变，我们研究发现不同拓扑系统的马约拉纳费米子遵从不同多边形法则，分别是拓扑多边形符号法则和一般多边形符号法则。我们设计了数值方法来区分马约拉纳费米子的多边形符号法则，通过计算发现对于二维系统，涡旋诱导的马约拉纳费米子遵从拓扑多边形法则，线位错诱导的马约拉纳费米子遵从一般多边形符号法则。