Exact model for extremely correlated electrons in a magnetic field

非技术总结相互作用电子的研究对于理解金属、半导体和超导体的性质至关重要，这些性质构成了当今技术的基础。该奖项支持理论和计算研究和教育，重点是相互作用的电子被限制在二维，“平地”，并与当它们被冷却到非常低的温度和暴露在高磁场时发生的现象。这种可以在两个半导体的界面上实现的选择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电阻的分量，称为霍尔电阻，电流的流动是电阻量子的倍数，它与材料无关，只取决于基本常数，如电子的电荷。宏观电阻变得量子化是出乎意料的;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，例如被称为量子威尔斯的半导体结构，单层和多层石墨烯，以及一类被称为过渡金属二硫属化物的材料，这些材料可能是石墨烯在某些技术应用中的替代品。这些系统的另一个方面是出现了新的和奇异的粒子类型，这些粒子在自然界的其他地方都没有发现，并且为未来的量子器件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将研究模型相互作用的后果，该模型相互作用是完全可解的，并表现出分数量子霍尔效应的观察到的现象学的几个方面。 这个模型与以前的模型不同的一个关键方面是，与动能相比，相互作用被认为是无限强的。该项目涉及调查完全可解的模型，以提供对现有概念的有用见解，并揭示富有成效的新的调查方向，以促进对局限于二维的电子的迷人特性的理解。分数量子霍尔效应将被研究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，以及提出的行为就像电子的一小部分的粒子。PI还计划探索对“平地”中电子之外的其他物理系统的推广。 该项目支持的学生将接受高级数值技术，分析场理论方法，拓扑概念和高性能计算的培训。他们还将参与国际合作。参加各种教育和外展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领域的了解，而且对研究生的专业成长至关重要，并将有助于培养他们在未来职业生涯中的基本技能。 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。 这种效应产生了一些新的概念，如分数电荷和分数统计粒子、复合费米子、拓扑超导、马约拉纳费米子和手征Luttinger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。对于某些分数量子霍尔波函数，精确的父哈密顿算子已经被构造出来，最早的例子是Halfton模型，它获得了Laughlin波函数作为精确的基态。这些模型假设回旋能量与相互作用能量相比是无限大的，因此电子被限制在最低的朗道能级。该项目的主要目标将是研究最近开发的模型所开辟的几个方向，该模型考虑了与回旋能量相比无限强的相互作用，并且可以解决所有本征态和本征能量。各种拓扑功能将明确评估从确切的波函数，这有一个相当复杂的形式。特别是，纠缠谱和辫统计的准粒子将被评估。波函数将被推广到球面和环面几何，这将带来他们的拓扑性质。配对不稳定性和非阿贝尔统计的物理学将在这个模型中得到解决，这也将被推广到玻色子，自旋电子和双层系统的分数量子霍尔效应，并包括部分子结构的启发。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外展活动。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.