分数量子霍尔效应拓扑序的研究

This project focuses on the foundamental theory in condensed matter physics. We try to understand the topological properties and the low-lying excitation properties in the fractional quantum Hall(FQH) states which are shown in new plateaus in semiconductors or new materials experimentally. The research contains searhing for the new FQH states, especially the non-Abelian FQH states in the two-dimensional electron gas(2DEG) system such as the GaAs/GaAlAs heterostructure, quantum well, or graphene system. We will explore the topological properties of the FQH states by studying their edge excitations and the quasiparticle excitations. In realistic situation, we should also consider the effects of disorder, the layer thickness of the 2DEG, anisotropy in system and Landau level mixing et al. Understanding the topological order and topological phase is a very hot and challenging topic in the condensed matter physics since it is out of the description of the Laudau's theory. Some of the FQH states, such as at the filling 5/2 or 12/5, have non-trivial quasiparticle excitation which satisfy non-Abelian statistics. The non-Abelian quasiparticles are crucial in the topological protected quantum computation. In this project, we study the FQH states by the help of large-scale numerical computations, simulations, and analytical calculations. We will try to implement the density matrix renormalization group (DMRG) method in the study of FQHE for large system sizes. This project should be helpful to understand the experimental results in recent years and also meaningful to the development of the fundamental theories in condensed matter physics.

本项目着眼于凝聚态物理的基础理论，主要研究分数量子霍尔效应中新的量子态的拓性质以及其低能激发性质。具体研究内容包括对半导体材料以及石墨烯材料等二维电子气系统中可能实现的分数量子霍尔态的拓扑性质以及拓扑相进行研究，尤其是对非阿贝尔量子霍尔态及其准粒子激发和边缘态激发的性质进行研究，并讨论在真实情况下系统中的杂质，二维电子气的厚度，各向异性以及朗道能级混合的影响。本项目注重理论和实验的结合，关于拓扑量子态的研究属于目前凝聚态物理中的前沿领域，因为它超出了利用对称性，序参量来描述量子态和量子相变的传统方法。主要结合大规模数值计算，模拟以及解析分析的方法来研究分数霍尔效应，并致力于发展密度矩阵重整化群等方法在分数量子霍尔效应研究上的应用。本项目有助于理解目前新的实验结果，而对具有拓扑序的物质，尤其是对具有非阿贝尔拓扑序的量子态的研究对于凝聚态物理理论的发展也具有较大意义。

本项目主要研究内容包括对半导体材料以及石墨烯材料等二维电子气系统中可能实现的分数量子霍尔态的拓扑性质以及拓扑相进行研究，尤其是对非阿贝尔量子霍尔态，如5/2 Moore-Read态和12/5 Read-Rezayi态的波函数及其准粒子激发和边缘态激发的性质进行研究，并讨论在真实情况下系统中的杂质，二维电子气的厚度，各向异性的影响。主要结合数值对角化计算，模拟以及解析分析的方法来研究分数霍尔效应，并致力于发展密度矩阵重整化群等方法在分数量子霍尔效应研究上的应用。我们通过对称性研究提出了形变石墨烯在磁场中的反常量子霍尔效应的理论，有助于理解相关实验结果。利用 Jack 多项式在较大系统中构造了基态，准粒子激发态，和边缘激发态的波函数，对非阿贝尔分数量子霍尔态和较为复杂的阿贝尔分数霍尔态的准粒子隧穿，边缘态激发以及重构，纠缠谱/熵等相关性质进行了研究，对5/2和12/5态的相干长度和相干温度的精确计算为干涉实验提供了参考数据，从而对分数量子霍尔效应的拓扑性质的理解得到了进一步深入。实现了利用 DMRG 算法在大尺度系统中的应用，对下一步深入研究打下良好的基础。另外我们提出了描述各向异性相互作用的赝势表示方法用于描述旋转对称性破缺的量子霍尔态，对对近年来新发现的二维材料，如 ZnO，black phosphorus 等可能实现的分数量子霍尔态的理解打下了良好的基础。