RUI: Theoretical (Numerical) Investigations of Novel Transport and Topological Properties of Two-Dimensional Interacting Electron Systems

该项目涉及理论（数值）方法来研究新出现的量子相的基本性质以及在几个电子系统中相关的新传输和拓扑性质。研究将在一个以本科生为主的机构进行，并将为本科生和研究生提供参与研究的机会。这项研究由三个项目组成。首先，首席研究员（PI）提出在具有强电子-电子相互作用的耦合双层电子系统中研究新出现的量子霍尔效应（QHE）。最近，通过在双层体系中传导逆流，实验建立了一个以整数QHE和激子凝聚共存为特征的新量子相。实验还提出了包括非阿贝尔配对霍尔态在内的新型分数阶QHE态。在理论方面，如何用电荷（总）霍尔电导以外的特征来表征量子霍尔态仍然是一个悬而未决的问题。为此，PI及其合作者开发了一种基于拓扑不变陈恩数矩阵的新数值方法，该方法将应用于研究逆流电流测量中的库仑拖动输运、电荷霍尔效应和输运性质以及双层系统中的量子相变。我们的目标是理解现有的实验观察，表征这些新量子态的量子相变的本质，并对未来实验的输运测量进行定量预测。其次，PI提出了石墨烯中二维相互作用狄拉克费米子的新型QHE。在最近的实验中，我们观察到单原子厚石墨（石墨烯）体系中狄拉克费米子出现了一系列具有非常规量子化规则的QHE平台。到目前为止，新能带结构、无序势和库仑相互作用的相互作用尚未得到研究，这可能很快成为量子he领域的中心课题之一。考虑到材料的所有这些重要方面，我们建议使用微观带模型进行系统的数值研究，这可能提供有价值的信息并进一步刺激实验研究。第三，对二维电子系统中电子-电子相互作用能否导致具有拓扑有序和分数化的自旋-液态这一基本问题也进行了数值研究。了解这一问题将对强相关电子系统理论和拓扑量子计算的未来发展产生重要影响。我们的目标是在低能谱、拓扑简并和自旋-自旋相关函数的广泛数值计算的基础上，确定二维电子系统中拓扑有序自旋-液态的一些具体例子。这一建议的智力价值在于，所讨论的主题对于理解二维电子系统中的新物理现象具有根本的重要性。拟议的研究项目对社会的广泛影响是双重的。首先，该项目将对未来新型磁电子器件和拓扑量子计算量子比特的发展产生影响。其次，本项目将为学生和博士后提供关于如何在物理学前沿开展研究的良好介绍和培训，并为他们在未来的学术和非学术生涯中处理实际问题做好准备。在过去的几年里，PI和她的合作者开发了新颖有效的数值方法，基于拓扑不变量，研究相互作用电子系统的量子输运和拓扑性质。因此，我们认为所提出的研究可以有效和成功地进行。非技术摘要：本项目涉及理论（数值）方法来研究新出现的量子相的基本性质，以及在固体中发现的几种电子系统中相关的新输运和拓扑性质。研究将在一个以本科生为主的机构进行，并将为本科生和研究生提供参与研究的机会。

This project involves theoretical (numerical) approaches to the fundamental nature of new emerging quantum phases and associated novel transport and topological properties in several electron systems. The research will be done at a predominantly undergraduate institution and will afford opportunities for undergraduates and graduate students to participate in the research. The research is comprised of three projects.Firstly, the Principal Investigator (PI) proposes to study the new emerging quantum Hall effect (QHE) in coupled bi-layer electron systems with strong electron-electron interaction. A new quantum phase characterized by the coexistence of integer QHE and exciton condensation has recently been established experimentally, by conducting counter-flow currents in bi-layer systems. Novel fractional QHE states including the non-Abelian paired Hall states are also suggested by experiments. On the theoretical side, it remains an open issue how to characterize quantum Hall states with features besides their charge (total) Hall conductance. A new numerical method based on a matrix of topological invariant Chern numbers has been developed by the PI and her collaborators for this purpose, which will be applied to study the Coulomb drag transport, charge Hall effect and the transport properties in counter-flow current measurement as well as quantum phase transitions in bi-layer systems. We are aiming at understanding the existing experimental observations, characterizing the nature of quantum phase transitions for these new quantum states, and making quantitative predictions regarding transport measurements for future experiments.Secondly, the PI proposes to study the novel QHE of two-dimensional (2D) interacting Dirac fermions in graphene. In recent experiments, a series of QHE plateaus with unconventional quantization rule have been observed for Dirac fermions in single-atom-thick graphite (graphene) system. So far, the interplay of the novel band structure, disorder potential and Coulomb interaction has not been studied yet, which may soon become one of the central topics in the field of QHE. We propose to perform a systematic numerical study using microscopic band model taking into account all these important aspects of the material, which may provide valuable information and further stimulate experimental research.Thirdly, the fundamental problem whether electron-electron interaction can lead to spin-liquid state with topological ordering and fractionalization in 2D electron system will also be investigated numerically. Understanding this issue will have important impact on the theory of strongly correlated electron systems and the future development of the topological quantum computing. We are aiming to identify some concrete examples of topological ordered spin-liquid state in 2D electron systems based on extensive numerical calculations of low energy spectrum, topological degeneracy and spin-spin correlation function.The intellectual merit of this proposal is that the topics addressed are of fundamental importance for the understanding of the new physical phenomena in 2D electron systems. The broader impact on society of the proposed research project is twofold. Firstly, the project will impact on the future development of new magneto-electronic devices and topological quantum computing qubits. Secondly, the present project will provide students and postdoctoral fellows with excellent introduction and training about how to carry out research at the forefront of physics and will also prepare them for dealing with practical problems in future academic and non-academic careers. In the past a few years, PI and her collaborators have developed novel and effective numerical methods, based on topological invariant quantities, to study the quantum transport and topological properties of interacting electron systems. Thus we believe that the proposed research can be carried out effectively and successfully.Non-Technical Abstract: This project involves theoretical (numerical) approaches to the fundamental nature of new emerging quantum phases and associated novel transport and topological properties in several electron systems that are found in solids. The research will be done at a predominantly undergraduate institution and will afford opportunities for undergraduates and graduate students to participate in the research.