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

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，这在室温和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，它发生在二维电子片中，通常在半导体之间的人工工程界面上，冷却到离绝对零度不到一度的温度，并置于垂直于电子片的强磁场中。在最简单的这种状态下，电子可以被认为是“分裂”成三个被称为复合费米子的物体。每个复合费米子的电荷是电子的三分之一，并具有其他奇异的性质。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条件。在过去的十年里，人们提出了一种潜在实现这种状态的新方法，即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，即拓扑能带材料。PI将调查拓扑能带材料是否可以支持以前未发现的状态，甚至在传统的量子霍尔系统中。这项研究的一个主要目标是以一种近似的分析方式来理解这些状态的性质，其中潜在的物理是清楚的。这将有助于理解传统量子霍尔态与拓扑能带材料之间的异同。PI还将利用这种方法研究受弹性应变影响的传统量子霍尔材料。任何真实的材料都包含晶格缺陷、取代原子和缺陷，这些缺陷统称为无序。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，无序的影响是重要的。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种学习许多学科的机制，并从实验学习中受益。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新型绝缘体，称为拓扑绝缘体。当这些带满时，它们具有量子化的霍尔电导。部分填充和强电子-电子相互作用形成分数量子类霍尔态。这项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)在主分数和集体激励下的间隙态的基态能量将用为分数量子霍尔效应开发的哈密顿方法计算。(b)利用基态能量交叉研究不同自旋主分数态之间的跃迁。(c)将研究半填充态的两种不同可能性，即电子流体和复合费米子流体。研究了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状态将使用守恒近似来研究。这与确定拓扑带材料是否具有传统分数量子霍尔态以外的激发有关。(e)拓扑带的两个时间反转副本是时间反转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，倾斜的场或应变产生了已测量的各向异性。PI将发展这种各向异性状态的分析理论，以及向列状状态的潜在相变。2. 阐明猝灭无序在量子霍尔铁磁体中的作用：典型系统是填充双层，实验观察对理论提出了许多挑战，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摄动效应。Hartree-Fock理论和有效的低能理论将用于确定响应于电势的拓扑电荷的产生。(b)将计算集体激发，以得出这种状态的光散射的实验特征。(c)将在不同拓扑电荷排列之间的相变附近建立低能场理论。(d)在这一阶段将引入弱无序，利用重整化群技术确定跃迁附近的低能量长波行为。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种学习许多学科的机制，并从实验学习中受益。

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.