Dynamics, Ground States, and Elementary Excitations of Quantum Many-Body Systems

电子设备的小型化一直是计算机不断增长的能力的驱动力，无论是表示为每单位体积的性能单位，还是每单位能量消耗，还是每美元花费。当电子设备中单个元件的尺寸接近单个原子的大小时，小型化将达到极限。除非我们成功地实现一种新的计算范式，称为量子计算，否则计算效率的增长，通常被称为摩尔定律，将走到尽头。实现量子计算思想的最有前途的研究方向之一是拓扑材料的发展。拓扑材料具有存储量子信息的能力，在普通条件下，由于一种称为退相干的过程，量子信息会迅速退化。在这个项目中，PI和他的合作者将研究拓扑材料的数学模型。主要目标是更好地理解决定基于拓扑材料的量子存储器鲁棒性的因素。该项目将解决三个问题，这些问题对于拓扑有序材料在可靠量子计算设备开发中的可能应用至关重要。首先，当系统尺寸增加时，什么决定了临界点处的谱隙消失的速率？第二，在什么条件下具有拓扑有序基态的系统的任意子激发谱在足够小的扰动下是稳定的？第三，随机性（如掺杂材料中发生的随机性）对基态结构和这种系统的低激发态有什么影响？该项目还将研究模型系统的动态行为。关于基态以上的低激发能量的问题，即，光谱间隙是量子多体物理学中许多问题的核心。该项目的目标是显着扩展方法的适用范围，以估计至少两个方向的频谱间隙。一种是差距随系统尺寸（临界点或临界区域）增大而消失的情况。第二个目标是开发分析二维或多维系统差距的方法，迄今为止仅在少数特殊情况下实现了这一目标。此外，在无序系统的情况下，频谱间隙可能会关闭，但所谓的迁移率间隙可能会发挥非常相似的作用，该项目的另一个目标是将当前的方法扩展到频谱间隙被迁移率间隙取代的情况。从表示理论，分析和概率的技术将被使用时，将在需要时开发新的方法。

Miniaturization of electronics has been the driving force of the ever increasing power of computers, whether expressed as units of performance per unit of volume, or per unit of energy consumed, or per dollar expended. Miniaturization will reach its ultimate limit when the size of individual components in electronic devices approaches the size of a single atom. Unless we succeed in implementing a new paradigm of computation, called Quantum Computation, the growth in computational efficiency, which is often referred to as Moore's Law, will come to an end. One of the most promising research directions to implement the ideas of Quantum Computation is the development of topological materials. Topological materials have the capacity to store quantum information, which under ordinary conditions is subject to fast deterioration due to a process called decoherence. In this project the PI and his collaborators will study mathematical models of topological materials. The main goal is to better understand the factors that determine the robustness of a quantum memory based on topological materials.The project will address three questions that are crucial for the possible application of topologically ordered materials to the development of reliable quantum computation devices. First, what determines the rate with which the spectral gap at critical points vanishes as the system size increases? Second, under what conditions is the anyonic excitation spectrum of systems with topologically ordered ground states stable under sufficiently small perturbations? Third, what is the effect of randomness (such as occurs in doped materials) on the structure of the ground state and low-lying excitations of such systems? The project will also investigate the dynamical behavior of the model systems. Questions regarding the energy of low-lying excitations above the ground state, i.e., the spectral gap, are central to many issues in quantum many-body physics. The goal of the project is to significantly extend the range of applicability of methods to estimate the spectral gap in at least two directions. One is the situation where the gap vanishes with increasing system size (critical points or regions). The second aim is to develop methods to analyze the gap of systems in two or more dimensions, which has so far only been achieved in a few special cases. Further, in the case of disordered systems the spectral gap may close, but a so-called mobility gap may play a very similar role and another goal of the project is to extend current methods to situations where the spectral gap is replaced by a mobility gap. Techniques from representation theory, analysis, and probability will be used when available and new methods will be developed when needed.