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

电子产品的小型化一直是计算机功率不断增加的驱动力，无论是以单位体积的性能单位表示的，还是以单位消耗的能量表示的，还是以消耗的美元表示的。当电子设备中的单个元件的尺寸接近单个原子的尺寸时，小型化将达到极限。除非我们成功地实现一种名为量子计算的新计算范式，否则计算效率的增长--通常被称为摩尔定律--将会结束。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，但在正常情况下，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。该项目的主要目标是更好地了解决定基于拓扑材料的量子存储器的健壮性的因素。该项目将解决三个问题，这三个问题对于可能将拓扑有序材料应用于开发可靠的量子计算设备至关重要。首先，是什么决定了临界点的光谱间隙随着系统规模的增加而消失的速度？其次，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(如在掺杂材料中发生的)对这类系统的基态结构和低激发态结构有什么影响？该项目还将调查模型系统的动力学行为。有关基态以上低激发能量的问题，即光谱能隙，是量子多体物理中许多问题的核心。该项目的目标是显著扩大估计至少在两个方向上的光谱差距的方法的适用范围。一种情况是，差距随着系统规模(关键点或区域)的增加而消失。第二个目标是开发在两个或更多维度上分析系统差距的方法，到目前为止，这只在少数几个特殊情况下实现。此外，在无序系统的情况下，频谱间隙可能会缩小，但所谓的迁移率间隙可能扮演非常类似的角色，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。当可用的时候，将使用表示理论、分析和概率的技术，当需要时，将开发新的方法。

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.