Exact model for extremely correlated electrons in a magnetic field

研究相互作用的电子对理解金属、半导体和超导体的特性至关重要，而这些特性是当今技术的基础。该奖项支持理论和计算研究和教育，重点是局限于二维“平面”的相互作用电子，以及当它们冷却到非常低的温度并暴露在高磁场中时发生的现象。这种可以在两个半导体的界面上实现的选举系统已知会产生非常不寻常的物质量子力学相，最值得注意的是那些被称为霍尔电阻的电阻组成部分，对电流的流动是一个量子电阻的倍数，这与材料无关，只取决于基本常数，如电子的电荷。宏观电阻被量子化是意料之外的；这种现象被称为分数量子霍尔效应。它已经成为二维量子材料中出现的新现象的跳板，例如被称为量子阱的半导体结构，单层和多层石墨烯，以及一类被称为过渡金属二硫族化合物的材料，这些材料可能在某些技术应用中替代石墨烯。这些系统的另一个方面是出现了新的和奇特类型的粒子，这些粒子在自然界的其他地方找不到，并且为未来的量子设备技术产生了创新的想法。为了进一步深入了解分数霍尔效应，PI将研究一个模型相互作用的结果，该模型相互作用是完全可解的，并展示了观察到的分数量子霍尔效应的几个方面。该模型与以往模型的一个关键区别在于，与动能相比，相互作用被认为是无限强的。该项目涉及对精确可解模型的研究，为现有概念提供有用的见解，并揭示富有成效的新研究方向，以推进对二维电子迷人物理的理解。将被研究的分数量子霍尔效应的方面包括提出的与电子或光子根本不同的新粒子——光的量子，以及提出的行为好像是电子的一小部分的粒子。PI还计划探索“平面”中电子以外的其他物理系统的推广。本项目支持的学生将接受高级数值技术、分析领域理论方法、拓扑概念和高性能计算方面的培训。他们还将接触到国际合作。参加各种教育和推广活动将是他们培训的一个组成部分，因为它不仅促进了公众对STEM领域的理解，而且对研究生的专业成长至关重要，并将帮助他们培养在未来职业生涯中很好的基本技能。该奖项支持理论和计算研究和教育，以探索分数量子霍尔效应研究的新方向。这种效应产生了新的概念，如分数电荷和分数统计的粒子、复合费米子、拓扑超导、马约拉纳费米子和手性卢廷格液体。其中一些已经在其他平台上找到了家，并推动了与拓扑量子计算相关的想法。对于某些分数阶量子霍尔波函数，已经建立了精确的母哈密顿量，最早的例子是霍尔丹的模型，该模型获得了拉夫林波函数作为精确基态。这些模型假设回旋加速器能量与相互作用能量相比无限大，因此电子被限制在最低朗道能级。该项目的主要目标是研究最近开发的一个模型所开辟的几个方向，该模型考虑了与回旋加速器能量相比无限强的相互作用，并且对所有特征态和本征能量都是可解的。各种拓扑特征将从具有相当复杂形式的精确波函数中显式地评估。特别是，准粒子的纠缠谱和编织统计将被评估。波函数将扩展到球面和环面几何，这将显示出它们的拓扑特征。配对不稳定性和非阿贝尔统计的物理学将在该模型中讨论，该模型也将推广到玻色子、自旋电子和双层系统的分数量子霍尔效应，并包括受部分子构造启发的结构。一个重要的目标将是确定解决方案的哪些方面与实验室实验有关，以及是否出现迄今未知的结构。该项目支持的学生将采用先进的分析和计算技术，如陈-西蒙斯场论、精确对角化研究和量子蒙特卡罗方法。他们还将在宾夕法尼亚州立大学的适当夏令营以及州立大学的年度艺术节上为大学预科学生组织教育和拓展活动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

NONTECHNICAL SUMMARYThe study of interacting electrons is crucial to understanding of the properties of metals, semiconductors, and superconductors, which form the foundation for today’s technology. This award supports theoretical and computational research and education focused on interacting electrons confined to two-dimensions, “flatland,” and with phenomena that occur when they are cooled to very low temperatures and exposed to a high magnetic field. This system of elections which can be realized at the interface of two semiconductors is known to produce very unusual quantum mechanical phases of matter, most notably those for which a component of resistance, known as the Hall resistance, to the flow of electrical current is a multiple of a quantum of resistance that is independent of the material and depends only on fundamental constants like the charge of an electron. That this macroscopic resistance becomes quantized was unexpected; this phenomenon is called the fractional quantum Hall effect. It has served as the springboard for the advent of new phenomena that occur in two-dimensional quantum materials, such as semiconductor structures known as quantum wells, single and multi-layer graphene, and a class of materials known as transition metal dichalcogenides that may be an alternative to graphene for some technical applications. An additional aspect of these systems is the emergence of new and exotic types of particles, which are not found elsewhere in nature, and which have generated innovative ideas for future quantum device technologies. To gain further insight into the fractional Hall effect, the PI will investigate the consequences of a model interaction that is exactly solvable and exhibits several aspects of the observed phenomenology of the fractional quantum Hall effect. A crucial aspect in which this model deviates from the previous models is that the interaction is taken to be infinitely strong compared to the kinetic energy. This project involves the investigation of exactly solvable models to provide useful insights into existing concepts and to reveal fruitful new directions of inquiry to advance understanding of the fascinating physic of electrons confined to two dimensions. Aspects of the fractional quantum Hall effects that will be investigated include proposed new particles that are fundamentally different from the electron or photon - the quantum of light, and proposed particles that behave as if they were a fraction of an electron. The PI also plans to explore generalizations to other physical systems beyond electrons in “flatland.” The students supported by this project will be trained in advanced numerical techniques, analytical field theoretical methods, topological concepts, and high-performance computing. They will also be exposed to international collaboration. Participation in various education and outreach activities will be an integral part of their training because it not only promotes public understanding of the STEM fields but also is of utmost importance for the professional growth of the graduate students and will help build essential skills that will serve them well in their future careers. TECHNICAL SUMMARYThis award supports theoretical and computational research and education to investigate new directions in the study of the fractional quantum Hall effect. This effect has given rise to new concepts, such as particles with fractional charge and fractional statistics, composite fermions, topological superconductivity, Majorana fermions, and chiral Luttinger liquids. Some of these have found homes in other platforms and have driven ideas related to topological quantum computation. Exact parent Hamiltonians have been constructed for certain fractional quantum Hall wave functions, the earliest example being Haldane’s model that obtained the Laughlin wave function as the exact ground state. These models assume that the cyclotron energy is infinitely large compared to the interaction energy, so that electrons are confined to the lowest Landau level. The primary objective of this project will be to investigate several directions opened by a recently developed model that considers an interaction that is infinitely strong compared to the cyclotron energy and is solvable for all eigenstates and eigen-energies. Various topological features will be explicitly evaluated from the exact wave functions, which have a rather complex form. In particular, the entanglement spectrum and braid statistics of the quasiparticles will be evaluated. The wave functions will be extended to the spherical and torus geometries, which should bring out their topological character. The physics of pairing instability and of non-Abelian statistics will be addressed within this model, which will also be generalized to fractional quantum Hall effects of bosons, spinful electrons and bilayer systems, and to include structures inspired by the parton construction. An important goal will be to ascertain what aspects of the solution relate to the laboratory experiments and whether hitherto unknown structures emerge. The students supported by this project will employ advanced analytical as well as computational techniques, such as Chern-Simons field theory, exact diagonalization studies, and quantum Monte Carlo method. They will also organize education and outreach activities for pre-college students at appropriate summer camps at Penn State as well as at the annual Arts Festival at State College.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.