Entanglement and Scattering in 1d and 2d

非技术综述：该奖项支持基础理论研究和教育，旨在通过研究多体态的纠缠和动力学，促进我们对低维凝聚态理论的量子方面的掌握。许多最有趣的多体物理系统，如超导体和超流体，都涉及许多粒子之间的相互作用，这些相互作用受量子力学规则的支配。量子力学的精髓之一被称为纠缠，这是一种物理系统的各个部分可以相互关联的现象，也就是说，从某种意义上说，“知道”系统的另一个部分发生了什么，即对一个部分的测量将决定另一个部分的测量结果，即使这两个部分在物理上是分开的。当涉及到少量粒子时，人们很好地理解了纠缠，甚至已经在实验上证明了这一点。纠缠在多体系统中的作用更加复杂和微妙，目前正在紧张的研究中。一个相关的非常困难和重要的挑战与理解多体系统的动力学有关。事实上，我们研究物理系统的主要方法是通过检查它们在各种外部探测器(如外部磁场和电场)下的行为。了解这种过程的动力学有助于解释它们的组成部分的集体行为，这些集体行为是造成超导和超流等现象的原因。纠缠和动力学是自然交织在一起的，因为纠缠可能会影响动力学，而动力学可能会揭示纠缠。本项目将集中于这两种效应及其关系，从量子系统中纠缠的高度理论方面，到开发方法来分析涉及量子动力学的实际实验测量，如超导体上的x射线散射和磁系统中的中子散射实验。表征纠缠和动力学及其关系也可能具有潜在的长期好处，为控制量子系统提供关键。该项目提供了一个极好的机会来培训研究生，并向他们介绍这些前沿的物理问题。这项研究还将伴随着旨在促进科学思维的公开讲座。技术综述：该奖项支持二维量子系统各方面的基础理论研究。研究的主要领域将涉及纠缠和动力学。第一个焦点将是通过研究纠缠哈密顿量中的局域性来研究量子晶格系统中的局域性。这些是有效的哈密顿量，只描述了量子系统的一部分的状态。最近令人兴奋的发展表明，在用共形场理论描述的一大类系统中，纠缠哈密顿量可能具有相对简单的局域性质。这些系统将从一般场论的角度进行探索，特别强调费米子的特殊但至关重要的情况。将特别关注新的、原创的和具有变革意义的理论思想的发展。在第二部分，一些相同的方法，特别是那些处理费米子行列式的方法，也将被用来研究动力学问题。特别是，将开发分析共振x射线散射中必不可少的动态过程的方法。这类实验最近已经发展成为研究相关系统的强大研究工具，例如高温超导体。对这种测量的解释需要进行详细的分析，这可能有助于解开导致高温超导的一些可能的机制和成分。同时，我们还将研究量子涨落对低维自旋模型的影响，这些模型是由磁性中的受挫现象驱动的。该项目预计将吸引几个不同的科学界参与，范围从高能物理、数学到实验。组织公开讲座和对学生进行培训将是活动的一个组成部分。

NON-TECHNICAL SUMMARY: This award supports fundamental theoretical research and education aimed at advancing our grasp of quantum aspects of condensed matter theory at low dimensions through the study of entanglement and dynamics in many-body states. Many of the most interesting many-body physical systems, such as superconductors and superfluids, involve interactions between many particles, which are governed by the rules of quantum mechanics. One of the quintessential elements of quantum mechanics is called entanglement, the phenomenon that parts of a physical system can be correlated, i.e. "know" what goes on in another part of the system in the sense that a measurement on one will determine the outcome of a measurement on the other, even when the two parts become physically separated. Entanglement is well understood when a small number of particles is concerned, and has even been experimentally demonstrated. The role of entanglement in many-body systems is more complicated and subtle, and is currently under intense study. A related very difficult and important challenge is related to understanding the dynamics of many-body systems. Indeed, our main way of investigating physical systems is by examining their behavior under various external probes such as external magnetic and electric field. Understanding the dynamics of such processes can help explain the collective behaviors of their constituents that are responsible for phenomena such as superconductivity and superfluidity. Entanglement and dynamics are naturally intertwined, as entanglement may affect dynamics, and dynamics may reveal entanglement. The present project will concentrate on both of these effects and their relations, from highly theoretical aspects of entanglement in quantum systems, to the development of methods to analyze actual experimental measurements that involve quantum dynamics such as x-ray scattering on superconductors and neutron scattering experiments in magnetic systems. Characterizing entanglement and dynamics and their relations may also have a potential long-term benefit of providing the keys to controlling quantum systems. The project presents an excellent opportunity to train graduate students and introduce them to these cutting-edge physics problems. The research will also be accompanied by public lectures aimed at the promotion of scientific thinking. TECHNICAL SUMMARY: This award supports fundamental theoretical research on aspects of two-dimensional quantum systems. The main areas of research will involve entanglement and dynamics. The first focus topic will be the investigation of the nature of locality in a quantum lattice system through studying the nature of locality in entanglement Hamiltonians. These are effective Hamiltonians that describe the state of only a part of a quantum system. Recent exciting developments show that in a large class of systems described by conformal field theories, entanglement Hamiltonians may be of relatively simple, local, nature. These systems will be explored from a general field theory perspective, with particular emphasis on the special but crucially important case of fermions. Special attention will be given to the development of new, original, and transformative theoretical ideas. In the second part, some of the same methods, especially those dealing with fermionic determinants, will also be utilized to study dynamical problems. In particular, methods will be developed to analyze the dynamical process essential in resonant x-ray scattering. Such experiments have recently grown into a powerful investigative tool for the study of correlated systems, such as high-temperature superconductors. The interpretation of such measurements necessitates a detailed analysis, which may help disentangle some of the possible mechanisms and ingredients leading to high-temperature superconductivity. In parallel, the effect of quantum fluctuations on low-dimensional spin models motivated by frustration phenomena in magnetism will be studied. The project is expected to engage several different scientific communities, ranging from high-energy physics and mathematics to experiment. Organization of public lectures and training of students will be an integral part of the activity.