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和教育，以研究具有许多粒子的量子系统的行为，例如晶体固体中的电子。大量的电子及其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，强磁场使受限于二维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的工作目标是将理论凝聚态和量子信息论的工具结合起来，以了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步。量子计算处理的是包含许多量子比特的系统，为许多量子力学粒子组成的系统的物理提供了一个新的视角，这是对传统量子力学粒子的补充。PI将把新的量子计算启发的方法与标准的凝聚态技术相结合，以更好地了解凝聚态的格局。这一计划的一个特别好处是，PI研究的一些奇异凝聚态可能反过来应用于新的量子计算平台的设计。根据这一奖项，PI将在研究生获得博士学位的整个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，并帮助他们发展适当的书面和口头交流技能，以促进他们的研究成果的传播。技术摘要：该奖项支持理论和计算研究和教育，以对物质的拓扑量子相进行分类，包括那些不平衡的和那些受额外对称性保护的相。PI将结合凝聚态物理、量子场论和量子信息论的思想来了解这些量子多体系统。具体地说，PI将把量子场论的拓扑特征，如连续统有效作用中的拓扑项，与原则上可以从晶格哈密顿量中直接提取的拓扑不变量联系起来，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比特系统，自然会涉及到量子信息论的思想。一种量子信息思想是非平凡的量子细胞自动机，它是局部么正的浅深度电路的推广。PI将探索使用这种量子元胞自动机来解开奇异的、“超越上同调”的、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共形自举。这是一种数值技术，它通过对局部算符的维谱进行限制，来约束可能存在的共形场论。在前期工作的基础上，PI的团队将研究如何通过将相应的t Hooft反常合并到共形自举中来约束可能存在于拓扑相边界的共形场理论。更一般地，PI的目的将是确定如何将拓扑场论中出现的关于非局部算符的信息合并到共形自举中。此外，PI还将探索非平衡环境中的拓扑效应，例如最近引入的测量诱导的纠缠转变。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，一个呈现拓扑相变的双重统计-力学模型。PI计划探索这种二元性的更一般版本。在这个奖项下，PI将在研究生获得博士学位的整个过程中支持和指导他们。PI将帮助学生在凝聚态物理的相关领域进行教育，并帮助他们发展适当的书面和口头交流技能，以促进他们的研究成果的传播。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.