Gapped ground state phases of quantum lattice systems

下一代量子技术准备开发新的量子物质相，其性质与著名的量子材料表现出的超流性和超导电性截然不同。超导电性的大部分方面已经有了很好的数学模型，超导材料几十年来已经成功地应用于许多技术应用中。然而，这个项目中将研究的新相是用被称为任意子的类粒子实体来描述的，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完全不同的性质。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。任意子已经在实验中实现了分数量子霍尔材料，当然仍然有惊喜，以揭示在进一步的研究。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。这项研究正好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，并将获得与理论科学和应用科学领域的一系列研究人员互动的机会，并鼓励他们在科技公司实习，作为他们教育的一部分。因此，该项目将帮助劳动力为新兴的量子技术行业做好准备。在这个项目中要追求的具体研究问题主要有两个方向。1)哈密顿量中相互作用的微小修正下基态以上能隙的稳定性。特别是，PI将证明具有多个基态的系统的体能隙(远离边界)、晶格费米子模型以及不满足通常无挫折性条件的重要模型类的结果。放弃无挫折感的假设将促使我们开发新的技术。2)分数量子霍尔(FQH)系统的光谱特性。在他们最近为1/3填充案例所做工作的基础上，主要调查者和合作者将分析其他填充分数的情况。该项目还将研究FQH阶段的稳定性，并调查霍尔丹伪势模型的推导过程，超出现有的‘有限-N’处理。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.