Interplay of Topological Order and Symmetry In and Out of Equilibrium

非技术总结该奖项支持关于凝聚态系统的奇异低温状态的理论研究和教育。尽管量子力学在微观层面上描述了物质的行为，但量子力学效应在宏观层面上通常是不可见的。20世纪80年代，在研究量子霍尔效应时，人们发现了一类新的所谓的“拓扑有序”态。在量子霍尔效应中，电子被限制在两个半导体之间的平面内，并被置于垂直于该平面的高磁场中。量子霍尔态显示了在宏观尺度上可以感受到的量子效应。其中包括只携带一小部分电子电荷的粒子行为的出现，以及只允许电荷单向流动的通道。为了更好地了解电子在材料中可以实现什么样的新状态以及如何在实验中检测到它们，PI和合作者将结合理论凝聚态物理和量子信息理论领域的方法，量子信息理论领域由信息论、量子力学、数学和计算机科学的思想组成。量子信息论提供了一种有用的方法来思考涉及量子力学状态操纵的计算和信息传输方面的问题。该奖项还支持现代理论凝聚态物理前沿研究生的教育。此外，PI将继续开发一门研究生水平的课程，除了标准的凝聚物质场理论外，还将结合量子信息论的新材料。此外，这项工作可能会产生额外的积极影响，因为凝聚态的想法可能会潜在地导致量子信息和量子计算的进步。技术总结该奖项支持关于强关联多体量子系统中对称性和拓扑学的组合效应的理论研究和教育。这种强关联系统可以实现超出标准金兹堡-朗道-威尔逊对称破缺范式的物质的零温度相。这个项目的很大一部分将是对表现出内在拓扑有序、对称保护特征或两者兼而有之的物质的相进行分类。这将包括平衡零温度带隙量子相和平衡系统的多体局域化(MBL)。具体的研究领域包括：1)对费米子对称保护相进行分类，了解它们可能表现出的强关联的拓扑表面态；2)从交换投影子的格子哈密顿量中提取拓扑和对称保护相的普适性质，并了解这种交换投影子哈密顿量存在的条件；3)对周期驱动(Floket)MBL量子系统中的拓扑相进行分类。PI将使用分析工具，如量子场论和精确可解模型，来证明间隙相的存在，还将使用数学方法，如拓扑量子场论和代数拓扑学，来研究和潜在地排除假设的拓扑序模式。PI将开发新的量子信息理论工具来解决这些问题。这个项目有可能发展对平衡中和不平衡中的拓扑顺序的基本理解。特别地，非平衡MBL系统中的拓扑序的分类问题不一定服从于与平衡情形相同的场论技术，但可以用量子信息论方法来有效地解决。PI将在与量子信息社区内的合作者正在进行的工作的基础上开发这样的方法，最近的结果表明，这将对物理学和量子信息社区都有用。这一奖项反映了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。

NONTECHNICAL SUMMARYThis award supports theoretical research and education on exotic low temperature states of condensed matter systems. Although quantum mechanics describes the behavior of matter at the microscopic level, quantum mechanical effects are typically not visible at the macroscopic scale. In the 1980s a new class of so-called `topologically ordered' states was discovered in studying the quantum Hall effect in which electrons are confined in a plane between two semiconductors and placed in a high magnetic field perpendicular to the plane. The quantum Hall states show quantum effects that are felt at the macroscopic scale. These include the emergence of particle like behavior that carries only a fraction of the electron charge and channels which only allow charge to flow in one direction.To gain a better understanding of what kinds of new states of electrons can be realized in materials and how they can be detected experimentally, the PI and collaborators will combine methods from theoretical condensed matter physics and the field of quantum information theory which is comprised of ideas from information theory, quantum mechanics, mathematics, and computer science. Quantum information theory provides a useful way to think about aspect of computing and information transmission involving the manipulation of quantum mechanical states. This award also supports the education of a graduate student at the frontiers of modern theoretical condensed matter physics. Furthermore, the PI will continue to develop a graduate level course that incorporates new material from quantum information theory in addition to standard condensed matter field theory. In addition, the work may have an additional positive impact in that condensed matter ideas may potentially lead to advances in quantum information and quantum computation.TECHNICAL SUMMARYThis award supports theoretical research and education on the combined effects of symmetry and topology in strongly correlated many-body quantum systems. Such strongly correlated systems can realize zero-temperature phases of matter beyond the standard Ginzburg-Landau-Wilson symmetry breaking paradigm. These can have intrinsic topological order and support fractionalized `anyon' excitations, or they can be symmetry protected.A large portion of this project will be to classify phases of matter which exhibit intrinsic topological order, symmetry-protected features, or both. This will include both equilibrium zero temperature gapped quantum phases and many-body localized (MBL) out of equilibrium systems. Specific areas of focus include: 1) classifying fermionic symmetry protected phases and understanding the strongly correlated topological surface states they can exhibit, 2) extracting universal properties of topological and symmetry protected phases from commuting projector lattice Hamiltonians and understanding the conditions under which such commuting projector Hamiltonians exist, and 3) classifying topological phases in periodically driven (Floquet) MBL quantum systems. The PI will use analytical tools, such as quantum field theory and exactly solvable models, to prove the existence of gapped phases, and will also use mathematical methods, such as topological quantum field theory and algebraic topology, to study and potentially rule out putative patterns of topological order. The PI will develop new quantum-information theoretic tools to address these problems.This project has the potential to develop fundamental understanding of topological order in and out of equilibrium. In particular, the problem of classifying topological order in non-equilibrium MBL systems is not necessarily amenable to the same field theory techniques as is the equilibrium case, but can be usefully addressed with quantum information theory methods. The PI will build on ongoing work with collaborators within the quantum information community to develop such methods, which recent results indicate will be useful for both the physics and quantum information communities.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.