Entanglement and Scattering in 1d and 2d

非技术性总结：该奖项支持基础理论研究和教育，旨在通过研究多体状态中的纠缠和动力学，推进我们对低维凝聚态理论量子方面的把握。许多最有趣的多体物理系统，如超导体和超流体，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。量子力学的一个典型元素被称为纠缠，这是一种物理系统的部分可以相互关联的现象，即“知道”系统的另一部分发生了什么，在这个意义上，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分开。当涉及少量粒子时，纠缠被很好地理解，甚至已经被实验证明。纠缠在多体系统中的作用更为复杂和微妙，目前正在进行深入的研究。与此相关的一个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式是通过检查它们在各种外部探针（如外部磁场和电场）下的行为。理解这些过程的动力学可以帮助解释它们的成分的集体行为，这些行为是超导性和超流性等现象的原因。纠缠和动力学是自然交织在一起的，因为纠缠可能影响动力学，而动力学可能揭示纠缠。目前的项目将集中在这两个影响和它们的关系，从量子系统中纠缠的高度理论方面，到分析涉及量子动力学的实际实验测量的方法的发展，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提供控制量子系统的密钥的潜在长期利益。该项目提供了一个极好的机会，培养研究生，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。该奖项支持二维量子系统方面的基础理论研究。主要研究领域将涉及纠缠和动力学。第一个焦点是通过研究纠缠哈密顿量中的定域性来研究量子晶格系统中的定域性。这些是描述量子系统的一部分状态的有效哈密顿量。最近令人兴奋的发展表明，在共形场论描述的一大类系统中，纠缠哈密顿量可能是相对简单的，本地的，性质。这些系统将从一般场论的角度进行探讨，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，原创的和变革性的理论思想的发展。在第二部分中，一些相同的方法，特别是那些处理费米行列式，也将被用来研究动力学问题。特别是，将开发的方法来分析共振x射线散射中必不可少的动力学过程。这种实验最近已经发展成为研究相关系统（如高温超导体）的强大研究工具。对这些测量结果的解释需要进行详细的分析，这可能有助于解开一些导致高温超导性的可能机制和成分。与此同时，我们也将研究量子涨落对低维自旋模型的影响。该项目预计将吸引几个不同的科学团体，从高能物理和数学到实验。组织公开讲座和培训学生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.