Topology in many-body quantum systems in and out of equilibrium

非技术概述：该奖项支持理论和计算研究以及教育，以调查具有许多粒子的量子系统的行为，例如晶体固体中的电子。大量的电子和它们固有的量子力学性质可以协同作用，导致物质的新状态。其中包括超导体，它可以零电阻导电，以及量子霍尔态，在量子霍尔态中，强磁场使被限制在二维空间的电子执行紧密的圆形轨道，从而迫使净电流流向系统的边缘。PI工作的目标是结合理论凝聚态物质和量子信息理论的工具来理解量子物质的这些状态。在过去的几年里，量子计算推动了凝聚态理论的重大进展。量子计算处理包含许多量子比特的系统，为许多量子力学粒子系统的物理学提供了一个新的视角，这是对传统系统的补充。PI将结合新的量子计算启发方法和标准凝聚态技术，以更好地理解凝聚态相的景观。这个项目的一个特别好处是，PI研究的一些奇特的凝聚态相可能反过来应用于新的量子计算平台的设计。根据该奖项，PI将在研究生攻读博士学位的过程中提供支持和指导。PI将帮助学生在凝聚态物理的相关领域进行教育，并帮助他们培养适当的书面和口头沟通技巧，以促进他们的研究成果的传播。技术概述：该奖项支持理论和计算研究以及对物质拓扑量子相进行分类的教育，包括那些非平衡态和受额外对称性保护的量子相。PI将结合凝聚态物理、量子场论和量子信息论的思想来理解这些量子多体系统。具体来说，PI将连接量子场论的拓扑特征，例如连续有效作用中的拓扑项，以及原则上可以直接从晶格哈密顿量中提取的拓扑不变量，例如任意子或缺陷激发的编织统计。由于晶格量子多体系统本质上是一个多量子位系统，因此量子信息理论的思想自然会涉及到这项工作。一种量子信息思想是非平凡量子元胞自动机，它是局部酉元的浅深度电路的推广。PI将探索使用这种量子细胞自动机来解开奇异的、“超上同调”的、受对称保护的物质拓扑相的效用。PI计划使用的另一个工具是保形引导。这是一种限制可能存在的共形场论的数值技术，通过在局部算符的维谱上设置边界。在初步工作的基础上，PI的团队将研究如何通过将相应的't Hooft异常合并到共形自举中来约束可以存在于拓扑相边界的共形场理论。更一般地说，PI的目的是确定如何将拓扑场论中出现的非局部算子的信息合并到共形自举中。此外，PI还将探索非平衡设置中的拓扑效应，例如最近引入的测量诱导纠缠跃迁。PI团队的初步工作已经发现，对于这种转变的某个简单实例，一个显示拓扑相变的双重统计力学模型。PI计划探索这种对偶性的更多通用版本。根据该奖项，PI将在研究生攻读博士学位的过程中提供支持和指导。PI将帮助学生在凝聚态物理的相关领域进行教育，并帮助他们培养适当的书面和口头沟通技巧，以促进他们的研究成果的传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

NON-TECHNICAL SUMMARY:This award supports theoretical and computational research and education to investigate the behavior of quantum systems with many particles, such as electrons in a crystalline solid. The large number of electrons and their inherent quantum mechanical nature can act in concert leading to new states of matter. Among these are superconductors, which can conduct electricity with zero resistance, and the quantum Hall states, in which a strong magnetic field causes electrons confined to two dimensions to execute tight circular orbits, thereby forcing net electrical current flow to thei edges of the system. The goal of the PI’s work is to combine tools from theoretical condensed matter and quantum information theory to gain an understanding of these states of quantum matter. In the last few years, quantum computing has motivated significant progress in condensed matter theory. Quantum computing deals with systems that contain many quantum bits, providing a new perspective on the physics of systems of many quantum mechanical particles that is complementary to the traditional one. The PI will combine new quantum computing inspired methods with standard condensed matter techniques to gain a better understanding of the landscape of condensed matter phases. One particular benefit of this program is that some of the exotic condensed matter phases studied by the PI may in turn have applications to the design of new quantum computing platforms.Under this award, the PI will support and mentor graduate students throughout their progress to a PhD. The PI will help educate the students in the relevant areas of condensed matter physics, and help them develop proper written and oral communication skills to facilitate the dissemination of their research.TECHNICAL SUMMARY: This award supports theoretical and computational research and education to classify topological quantum phases of matter, including those that are out of equilibrium and those protected by additional symmetries. The PI will combine ideas from condensed matter physics, quantum field theory, and quantum information theory to gain an understanding of these quantum many-body systems. Specifically, the PI will connect the topological features of quantum field theories, such as topological terms in continuum effective actions, to topological invariants that can in principle be directly extracted from a lattice Hamiltonian, such as braiding statistics of anyon or defect excitations. Because a lattice quantum many-body system is essentially a many-qubit system, it is natural that ideas from quantum information theory will naturally be involved in this work. One quantum information idea is that of a non-trivial quantum cellular automaton, which is a generalization of a shallow-depth circuit of local unitaries. The PI will explore the utility of using such quantum cellular automatons to disentangle exotic, “beyond-cohomology,” symmetry protected topological phases of matter.An additional tool that the PI plans to use is that of the conformal bootstrap. This is a numerical technique that constrains the possible conformal field theories that can exist, by putting bounds on their spectra of dimensions of local operators. Building on preliminary work, the PI's team will research how to constrain the conformal field theories that can exist at the boundaries of topological phases, by incorporating the corresponding 't Hooft anomalies into the conformal bootstrap. More generally, the aim of the PI will be to determine how to incorporate information about the non-local operators that appear in topological field-theories into the conformal bootstrap. Furthermore, the PI will also explore topological effects in non-equilibrium settings, such as the recently introduced measurement-induced entanglement transition. Preliminary work by the PI's team has uncovered, for a certain simple instance of this transition, a dual statistical-mechanics model which exhibits a topological phase transition. The PI plans to explore more general versions of this duality.Under this award, the PI will support and mentor graduate students throughout their progress to a PhD. The PI will help educate the students in the relevant areas of condensed matter physics and help them develop proper written and oral communication skills to facilitate the dissemination of their research.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.