On locality and randomness in non-relativistic systems

该项目的研究者：“非相对论系统中的局部性和随机性”将集中在两个主要的感兴趣的主题。第一部分探索了与非相对论系统相对应的动力学准局部结构。由于这样的系统不具有与有限光速等价的事实，与例如最近邻哈密顿量相关联的动力学在严格意义上通常不保持局部性。局域性或准局域性的一种近似形式，已被证明是许多系统所支配的哈密顿与短程相互作用。最近的发展表明，这些估计，称为Lieb-Robinson界限，在各种情况下都是有效的，并在一些重要的应用程序中是有用的。研究人员提出了几个系统的研究，没有已知的准局域性的结果。鉴于这些估计，某些有趣的问题可能是在严格和详细的分析范围内。下一个焦点领域是关于特定量子自旋系统的随机扰动。对于单粒子系统，众所周知，无序导致局域化：具有足够多杂质的金属失去其导电特性。这种现象应该在多粒子相互作用系统中持续存在，研究人员建议分析随机量子自旋系统中局域化的出现。事实上，一个动力学形式的本地化，在明确的Lieb-Robinson边界，被提出作为一种可能的手段，量化，从而进一步调查这一预测behavior.In凝聚态物理，丰富的有趣的，物理相关的现象被描述在量子多体系统的背景下。例如，晶体中原子的振荡和磁体中自旋的排列都可以用量子晶格模型非常有效地描述。这些量子晶格系统也为更多的理论考虑提供了一个框架。事实上，量子场论的离散版本以及量子信息和计算理论中的基本模型都可以在这种环境中表达。该提案的主要目标是通过分析一些特定模型来进一步发展量子晶格系统的理论。在这个建议中的一个重要组成部分将是调查的地方估计表明，干扰不传播任意快速通过所考虑的系统。这些定域性界限的几个重要应用已经出现在数学和物理文献中。作为一个新的例子，研究者提出了一个详细的分析随机系统的局部性。随机系统是特别有趣的，因为它们被预测表现出金属-绝缘体相变，随着无序度的增加。足够详细的分析可能会建议材料的设计规范，例如光纤电缆，这些材料必须分散在各处。在这种情况下，这一提议的潜在影响，超出了数学和物理学的范围，可能是巨大的。此外，对该项目感兴趣的研究生将学习科学家在许多领域使用的语言和技术。完成论文后，他们应该为科学的学术职位或工业中更具技术性的角色做好充分准备。

The investigator of the project: "On locality and randomness in non-relativistic systems" will focus on two major topics of interest. The first explores a quasi-local structure of the dynamics corresponding to non-relativistic systems. Due to the fact that such systems have no equivalent to a finite speed of light, the dynamics associated with, for example, a nearest neighbor Hamiltonian does not generally preserve locality in the strict sense. An approximate form of locality, or quasi-locality, has been demonstrated for many systems governed by Hamiltonians with short-range interactions. Recent developments have shown that these estimates, known as Lieb-Robinson bounds, are valid in a variety of contexts and useful in a number of important applications. The investigator proposes the study of several systems for which there are no known quasi-locality results. Given these estimates, certain intriguing questions may be within reach of rigorous and detailed analysis. The next area of focus concerns random perturbations of specific quantum spin systems. For single particle systems, it is well known that disorder leads to localization: a metal with sufficiently many impurities loses its conductance properties. This phenomenon should persist in many-particle interacting systems, and the investigator proposes to analyze the emergence of localization in random quantum spin systems. In fact, a dynamical form of localization, in terms of explicit Lieb-Robinson bounds, is proposed as a possible means of quantifying and thus further investigating this predicted behavior.In condensed matter physics, a wealth of intriguing, physically relevant phenomena is described in the context of quantum many-body systems. For example, the oscillations of atoms in a crystal and the alignment of spins in a magnet can both be quite effectively described by quantum lattice models. These quantum lattice systems also provide a framework for more theoretical considerations. In fact, discrete versions of quantum field theory and the basic models in the theory of quantum information and computation can be expressed in this setting. The main goal of this proposal is to further develop the theory of quantum lattice systems by analyzing a number of specific models. A crucial ingredient in this proposal will be the investigation of locality estimates which demonstrate that disturbances do not propagate arbitrarily fast through the systems under consideration. Several important applications of these locality bounds have already appeared in the mathematics and physics literature. As a new example, the investigator proposes a detailed analysis of locality properties for random systems. Random systems are particularly interesting because they are predicted to exhibit a metal-insulator phase transition with increasing disorder. Sufficiently detailed analysis may suggest design specifications for materials, e.g. fiber-optic cables, that necessarily have impurities dispersed throughout. In this case, the potential impact of this proposal, beyond the scope of mathematics and physics, may be substantial. Moreover, interested graduate students working on the project will learn the language and techniques used by scientists in a number of fields. After completing their thesis, they should be well prepared for either an academic position in the sciences or a more technical role in industry.