Gapped ground state phases of quantum lattice systems

下一代量子技术准备开发物质的新量子相，其特性与众所周知的量子材料表现出超流动性和超导性完全不同。几十年来，超导材料已经成功地应用于许多技术领域。然而，在这个项目中将要研究的新相是用被称为任意子的类粒子实体来描述的，它的性质与描述传统材料（包括超流体和超导体）的传统玻色子和费米子截然不同。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被实验实现，并且在进一步的研究中肯定仍有惊喜有待揭示。在这个项目中，首席研究员（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。这项研究完全符合美国国家科学基金会的“量子飞跃”大构想。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互动，并鼓励他们在科技公司实习，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产业准备劳动力。本项目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变化下，基态以上间隙的稳定性。特别是，PI将证明具有多个基态系统的体隙（远离边界），晶格费米子模型以及不满足通常无挫折条件的重要模型类的结果。放弃无挫折的假设将促使我们开发新技术。分数量子霍尔（FQH）系统的光谱特性。基于他们最近对1/3填充情况的研究，首席研究员和合作者将分析其他填充部分的情况。该项目还将研究FQH相的稳定性，并在现有的“有限n”处理之外研究Haldane伪势模型的推导。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。

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.