Hamiltonian Theory of Fractionally Filled Chern Bands, and Disorder in Quantum Hall Ferromagnets

非技术总结该奖项支持旨在研究电子新状态的理论研究和教育。电子通常被认为是不可分割的，这在室温下以及在固态器件正常工作的条件下是正确的。对这项研究最重要的是量子霍尔态，它发生在一个二维的电子片中，通常在半导体之间的人工设计界面上，冷却到绝对零度不到一度的温度，并放置在垂直于片的非常强的磁场中。在最简单的这种状态下，电子可以被认为是“分裂”成三个物体，称为复合费米子。每个复合费米子的电荷是电子电荷的三分之一沿着，并具有其他奇特的性质。激发的电荷和其他性质取决于特定的状态。其中一些状态为量子计算提供了可能的平台。传统的半导体量子霍尔态需要极端的条件。在过去的十年中，人们提出了一种可能实现这种状态的新方法，其中某些材料内部的环境可以产生相当于非常强的内部磁场。PI的目标是研究传统的量子霍尔态和这些新发现的可能性，称为拓扑带材料。PI将研究拓扑能带材料是否可以支持以前没有发现的状态，即使是在传统的量子霍尔系统中。研究的一个主要目标是以一种近似的分析方式来理解这种状态的性质，其中底层物理是清楚的。这将有助于理解传统量子霍尔态和拓扑能带材料之间的相似性和差异。 PI还将利用这种方法来研究传统的量子霍尔材料受到弹性应变。任何真实的材料都包含晶格缺陷、取代原子和缺陷，统称为无序。PI将开发一种受控的方法来研究无序的作用，其中一些量子霍尔态的实验表明无序的影响很重要。PI还将为印度冬季学校的组织做出贡献，并将参与大学荣誉计划的重组，该计划为学生提供一种机制，让他们了解许多学科并从实验学习中受益。技术总结该奖项支持旨在研究材料中的量子霍尔态和拓扑态的理论研究和教育。最近已经确定，由于能带结构的拓扑性质，具有强自旋轨道耦合的材料可以形成称为拓扑绝缘体的新型绝缘体。当这样的频带是完整的，他们有一个量子化的霍尔电导。在部分填充和强的电子-电子相互作用下，形成分数量子类霍尔态。本研究有两个主要目的：1.研究分数填充拓扑带中的新态：PI将使用分析方法研究拓扑带中的复合费米子态。(a)基态能量的间隙状态的主要分数和集体激发将计算在哈密顿方法开发的分数量子霍尔效应。(b)不同自旋的主分数态之间的跃迁将使用基态能量交叉进行研究。(c)两种不同的可能性的半满状态，电子流体和复合费米子流体，将被调查。将研究相变的性质和相变附近的低能激发。(d)分数填充拓扑带的边缘态将使用守恒近似进行研究。这与确定拓扑能带材料是否具有不同于传统分数量子霍尔态的激发有关。(e)拓扑带的两个时间反演副本是时间反演不变拓扑绝缘体的模型。 在这个模型中将研究强相互作用电子的分数填充态。(f)在传统的分数量子霍尔效应中，倾斜场或应变产生已被测量的各向异性。PI将发展这种同位素状态的分析理论，以及向列相态的潜在相变。2.阐明量子霍尔铁磁体中淬灭无序的作用：原型系统是填充1双层，实验观察对理论提出了许多挑战，并且无序似乎是必不可少的。(a)PI和合作者将在第一阶段通过在量子霍尔系统上施加强周期势来模拟无序的非微扰效应。Hartree-Fock和有效低能理论将用于确定响应于电势的拓扑电荷的产生。(b)集体激发将被计算，以获得这种状态的光散射的实验签名。(c)在不同拓扑电荷排列之间的相变附近将构建低能场论。(d)在这一阶段，我们将引入弱无序，并利用重整化群技术来确定跃迁附近的低能长波长行为。PI还将为印度冬季学校的组织做出贡献，并将参与大学荣誉计划的重组，该计划为学生提供了一个了解许多学科并从实验学习中受益的机制。

NONTECHNICAL SUMMARYThis award supports theoretical research and education aimed to investigate novel states of electrons. Electrons are generally thought to be indivisible which is true at room temperature and under the conditions under which solid state devices normally operate. Most important to this research are the quantum Hall states, which occur in a two-dimensional sheet of electrons, usually at an artificially engineered interface between semiconductors, cooled to temperatures less than one degree from absolute zero, and placed in an a very strong magnetic field perpendicular to the sheet. In the simplest such state the electron can be thought of as "split" into three objects known as composite fermions. Each composite fermion has a charge one-third that of the electron along with other exotic properties. The charge and other properties of the excitation depend on the particular state. Some of these states offer possible platforms for quantum computing. Conventional, semiconductor quantum Hall states need extreme conditions. In the past decade a new way of potentially realizing such states has been proposed where the environment inside some materials can generate the equivalent of very strong internal magnetic fields. The PI aims to study both conventional quantum Hall states and these newly discovered possibilities, known as topological band materials. The PI will investigate whether topological band materials can support states that have not been discovered before, even in traditional quantum Hall systems. A main goal of the research is to understand the properties of such states in an approximate analytical way where the underlying physics is clear. This will enable understanding the similarities and the differences between conventional quantum Hall states and those in topological band materials. The PI will also utilize this approach to investigate conventional quantum Hall materials subjected to elastic strain. Any real material contains lattice imperfections, substituted atoms, and defects, collectively known as disorder. The PI will develop a controlled approach to investigate the role of disorder where experiments on some quantum Hall states suggest that the effect of disorder is important.The PI will also contribute to the organization of Winter Schools in India and will participate in the reorganization of the University Honors Program which provides a mechanism for students to learn about many disciplines and benefit from experimental learning.TECHNICAL SUMMARYThis award supports theoretical research and education aimed to investigate quantum Hall states and topological states in materials. It has recently been established that materials with strong spin-orbit coupling can form new types of insulators, known as topological insulators, because of the topological properties of the band structure. When such bands are full, they have a quantized Hall conductance. With partial filling and strong electron-electron interactions fractional quantum Hall-like states form. The research has two major thrusts: 1. Investigating novel states in fractionally filled topological bands: The PI will use an analytical approach to investigate Composite Fermion states in topological bands. (a) Ground state energies for gapped states at the principal fractions and collective excitations will be computed in the Hamiltonian approach developed for the fractional quantum Hall effect. (b) Transitions between principal fraction states of different spin will be investigated using ground state energy crossings. (c) Two different possibilities for the half-filled state, an electron fluid and a Composite Fermion fluid, will be investigated. The nature of the phase transition and low-energy excitations near the phase transition will be studied. (d) Edge states of fractionally filled topological bands will be studied using a conserving approximation. This is relevant for determining whether the topological band materials have excitations other than those of conventional fractional quantum Hall states. (e) Two time-reversed copies of topological bands are a model of a time-reversal invariant topological insulator. Fractionally filled states of strongly interacting electrons will be studied in this model. (f) Tilted fields or strain in the conventional fractional quantum Hall effects produces an anisotropy which has been measured. The PI will develop an analytical theory of such anisotopic states, and potential phase transitions into nematic-like states. 2. Elucidating the role of quenched disorder in quantum Hall ferromagnets: The prototypical system is the filling 1 bilayer, where experimental observations pose numerous challenges to theory, and where disorder seems to be essential. (a) The PI and collaborators will, at the first stage, mimic the nonperturbative effects of disorder by imposing a strong periodic potential on the quantum Hall system. Hartree-Fock and effective low-energy theories will be used to determine the generation of topological charges in response to the potential. (b) Collective excitations will be computed to derive an experimental signature in light scattering of such states. (c) A low-energy field theory will be constructed near the phase transitions between different arrangements of topological charge. (d) Weak disorder will be put in at this stage and renormalization group techniques will be used to determine the low-energy long-wavelength behavior near the transitions. The PI will also contribute to the organization of Winter Schools in India and will participate in the reorganization of the University Honors Program which provides a mechanism for students to learn about many disciplines and benefit from experimental learning.