Quasi-Locality Properties of Quantum Many-Body Dynamics and Applications

各种基本现象可以用数学模型表示为由许多相互作用的组成部分组成的系统。如果我们认识到单个组件主要只与少量“附近”组件交互，那么分析任何此类系统的动力学问题就变得容易处理了。这被称为系统的准局部性。在微观层面上，量子物理的基本定律支配着由原子和基本粒子组成的系统的动力学。最近在理解这类系统的动力学方面取得的许多进展都是通过对其准局域性的数学分析取得的。在这个项目中，利用量子系统的准局部性的新数学进展将被用来更好地理解新量子技术感兴趣的物理系统。特别是，将开发新的方法，以便对用于量子信息处理和计算的量子系统进行更精确的建模。低温下的量子多体系统描述了一系列迷人的行为，通常被描述为物质的量子态。这个项目将加深对所谓间隙相的数学理解。处于隙相中的系统具有一个或多个基态，并且在体中存在激发的谱隙。该项目的一个重要目标是了解如何将这些系统的激发谱描述为准粒子系统。特别是在两个空间维度的系统中，这些准粒子可能是任意子，而不是更常见的费米子和玻色子。基于动力学的基本准局部性，将开发技术来证明任意子的存在，它们与基态拓扑顺序的关系，以及它们的动力学性质。这些问题的动机是希望更多地了解凝聚态系统中物质的量子态，并更好地了解拓扑有序材料在量子存储器和其他量子器件等应用中的潜力。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。

A variety of fundamental phenomena can be modeled mathematically as systems of many interacting components. The problem of analyzing the dynamics of any such system becomes tractable if we recognize that individual components interact primarily with only a small number of "nearby" components. This is referred to as a quasi-locality property of the system. At the microscopic level, the basic laws of quantum physics govern the dynamics of systems consisting of atoms and elementary particles. Much of the recent progress made in understanding the dynamics of such systems has been through the mathematical analysis of their quasi-locality properties. In this project, new mathematical advances exploiting the quasi-locality properties of quantum systems will be used to obtain a better understanding of the physical systems of interest for new quantum technologies. In particular, new methods will be developed to allow more precise modeling of the quantum systems employed for quantum information processing and computation.Quantum many-body systems at low temperatures describe a fascinating array of behavior generically described as quantum states of matter. This project will deepen the mathematical understanding of so-called gapped phases. Systems in a gapped phase have one or more ground states and a spectral gap for excitations in the bulk. One important goal of the project is to understand how the excitation spectrum of such systems can be described as a system of quasi-particles. In particular for systems in two space dimensions these quasi-particles may be anyons, instead of the more commonly encountered fermions and bosons. Techniques will be developed, based on the fundamental quasi-locality properties of the dynamics, to prove the existence of anyons, their relation to topological order in the ground state(s), and their dynamical properties. These problems are motivated by the desire to learn more about the quantum states of matter that occur in condensed matter systems, and to understand better the potential of topologically ordered materials in applications such as quantum memory and other quantum devices.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.