On locality and randomness in non-relativistic systems

“非相对论系统中的局部性和随机性”项目的研究人员将专注于两个感兴趣的主要主题。第一种是探索与非相对论系统相对应的动力学的准局部结构。由于这样的系统没有与有限光速等价的事实，例如，与最近邻哈密顿量相关的动力学通常不保持严格意义上的局域性。对于许多由哈密顿控制的具有短程相互作用的系统，已经证明了定域性或准定域性的近似形式。最近的发展表明，这些被称为Lieb-Robinson界的估计在各种情况下都是有效的，并且在一些重要的应用中是有用的。研究者建议研究几个没有已知的准局域结果的系统。考虑到这些估计，某些耐人寻味的问题可能在严格而详细的分析范围之内。下一个焦点领域涉及特定量子自旋系统的随机微扰。对于单粒子系统，众所周知，无序导致局域化：含有足够多杂质的金属失去其电导性质。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，一种动力学形式的局域化，根据显式的Lieb-Robinson界限，被提出作为一种可能的方法来量化并进一步研究这种预测的行为。在凝聚态物理中，大量有趣的、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相当有效地描述。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个背景下表达出来。这一建议的主要目标是通过分析一些具体的模型来进一步发展量子晶格系统的理论。这项提议的一个关键组成部分将是对地点估计的调查，这些估计表明扰动不会在所考虑的系统中任意快速传播。这些局域界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，研究者提出了对随机系统局部性性质的详细分析。随机系统特别令人感兴趣，因为它们被预测呈现出无序增加的金属-绝缘体相变。足够详细的分析可能会提出材料的设计规范，例如光缆，这些材料必须具有分散在各处的杂质。在这种情况下，这一提议的潜在影响可能超出数学和物理的范围，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，他们应该做好充分准备，要么在科学领域从事学术工作，要么在工业领域担任更具技术性的职位。

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.