Gapped ground state phases of quantum lattice systems

下一代量子技术准备开发新的量子相物质，其性质与众所周知的量子材料表现出的超流性和超导性完全不同。超导性的大多数方面都存在良好的数学模型，超导材料已经成功地应用于许多技术应用几十年。然而，在这个项目中将要研究的新阶段是用称为任意子的粒子状实体来描述的，它具有与描述传统材料（包括超流体和超导体）的传统玻色子和费米子截然不同的性质。任意子是构建量子存储器和其他量子信息设备的潜在候选者。任意子已经在分数量子霍尔材料中得到了实验实现，并且在进一步的研究中肯定会有惊喜。在这个项目中，主要研究者（PI）和合作者将开发有效描述这些新材料所需的基本数学技术。这项研究福尔斯完全符合美国国家科学基金会的大创意“量子飞跃”。研究生和本科生将积极参与该项目，并将有机会与纯科学和应用科学领域的一系列研究人员进行互动，并鼓励他们在技术公司实习，作为他们教育的一部分。因此，该项目将有助于为新兴的量子技术行业准备劳动力。在这个项目中要解决的具体研究问题主要有两个方向。1)在哈密顿量中相互作用的微小修改下基态之上的差距的稳定性。特别是，PI将证明具有多个基态的系统的体隙（远离边界）的结果，晶格费米子模型，以及不满足通常的无挫折条件的重要类别的模型。放弃无挫折的假设将促使我们开发新的技术。 2)分数量子霍尔（Fractional Quantum Hall，简称FWHH）系统的光谱性质。基于他们最近对1/3填充病例的工作，主要研究者和合作者将分析其他填充分数的情况。该项目还将研究稳定性的氢阶段和调查的推导出Halcantine的赝势模型超出现有的'有限N'的治疗。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

The next generation of quantum technologies is poised to exploit new quantum phases of matter with properties quite unlike the well-known quantum materials exhibiting superfluidity and superconductivity. Good mathematical models for most aspects of superconductivity already exist, and superconducting materials have been successfully applied in many technological applications for decades. The new phases that will be studied in this project, however, are described in terms of particle-like entities called anyons, which have properties quite distinct from the traditional bosons and fermions that describe traditional materials (including superfluids and superconductors). Anyons are potential candidates to build quantum memory and other quantum information devices. Anyons have already been experimentally realized in fractional quantum Hall materials, and surely still hold surprises to be revealed in further studies. In this project, the principal investigator (PI) and collaborators will develop fundamental mathematical techniques required for the effective description of these new materials. This research falls squarely within NSF's Big Idea 'Quantum Leap'. Graduate and undergraduate students will be actively involved with the project and will be offered opportunities to interact with a range of researchers in pure and applied science and encouraged to pursue internships in technology companies as part of their education. Thus, the project will help prepare the workforce for the emerging quantum technology industry. Specific research problems to be pursued in this project fall in two main directions. 1) Stability of the gap above the ground state under small modifications of the interactions in the Hamiltonian. In particular, the PI will prove results for the bulk gap (away from boundaries) for systems with multiple ground states, for lattice fermion models, and for important classes of models that do not satisfy the usual frustration-freeness condition. To drop the frustration-freeness assumption will prompt us to develop new techniques. 2) Spectral properties of fractional quantum Hall (FQH) systems. Building on their recent work for the 1/3 filled case, the principal investigator and collaborators will analyze the situation of other filling fractions. The project will also study the stability of FQH phases and investigate the derivation of Haldane's pseudo-potential models beyond the existing 'finite-N' treatments.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.