Interactions and Topology in Quantum Matter: From frustrated magnets to fractional topological insulators

新类型的量子秩序和激发携带分数量子数预计出现在低能量尺度的广泛的凝聚态系统，包括二维超冷电子在强磁场（量子霍尔系统），和挫折的磁性材料与竞争的相互作用。这些领域之间的快速发展的接口，特别是相互作用的自旋轨道耦合材料的拓扑相，是这个建议的重点。除了具有根本意义外，这些系统可能是未来技术发展的关键，例如拓扑量子计算。这些系统的一个核心挑战是在量子力学水平上理解相互作用和拓扑结构之间的微妙相互作用。在受抑量子磁体中，拓扑通过非平凡的晶格结构表现出来，导致许多近简并态和由此产生的丰富而复杂的低能行为。正如人们最近才充分认识到的那样，自旋轨道耦合为受抑系统增添了新的方面，特别是包括虹膜。受最近和正在进行的实验活动的启发，该提案旨在更好地了解价键晶体和自旋液体之间的竞争，以及提供这些方案的可观察签名。拓扑结构也可以表现为非平凡能带结构的形式，如拓扑绝缘体的情况。除了早期的量子霍尔系统，在那里复杂的多体状态的形式作为一个后果的残余相互作用在巨大的简并朗道水平，迄今为止的重点是基本上非相互作用的实现。然而，一类新的高度相关的状态-分数拓扑（陈）绝缘体-最近被建议在晶格系统中形成，该晶格系统具有非平凡拓扑的几乎无色散带。这样的带可以在自旋轨道耦合的固态材料中实现，并且不需要外部磁场。原则上，这开辟了许多有趣的可能性，包括量子霍尔现象的晶格实现和室温下的非阿贝尔任意子。然而，这些系统提出了一些重要的新的理论挑战，由于相互作用，能带拓扑结构和底层晶格的综合影响而出现。理解这种相互作用，最明显地表现为不同的贝里曲率，是这个建议的一个主要目标。为了研究这些系统，我们建议使用基于纠缠的模拟方法和诊断方法，以及更传统的凝聚态理论方法，包括级数展开，精确对角化和能带结构考虑。

New types of quantum order and excitations carrying fractional quantum numbers are predicted to emerge at low energy scales in a wide spectrum of condensed matter systems including twodimensional ultra cold electrons in a strong magnetic field (the quantum Hall system), and frustrated magnetic materials with competing interactions. The rapidly developing interface between these fields, in particular interacting topological phases in spin-orbit coupled materials, is the focus of this proposal. In addition to being of fundamental interest, these systems may hold the key to future technological developments such as topological quantum computation. A central challenge in these systems is to understand the subtle interplay between interactions and topology at the quantum mechanical level. In frustrated quantum magnets, topology is manifested through the non-trivial lattice structure, leading to many nearly degenerate states and a resulting rich and complicated low-energy behavior. As has been fully appreciated only recently, spin-orbit coupling adds novel aspects to frustrated systems, notably including the iridates. Motivated by intriguing recent and ongoing experimental activities, this proposal aims at a better understanding of the competition between valence bond crystals and spin liquids, as well as providing observable signatures of these scenarios. The topology can also be manifested in the form of a non-trivial band structure as in the case of topological insulators. With the early exception of the quantum Hall system, where complex manybody states form as a consequence of residual interactions within hugely degenerate Landau levels, the focus has so far been on essentially non-interacting realizations. However, a new class of highly correlated states - fractional topological (Chern) insulators - has very recently been suggested to be form in lattice systems harboring nearly dispersionless bands with non-trivial topology. Such bands may be realized in spin-orbit coupled solid-state materials and do not require an external magnetic field. In principle, this opens up a number of intriguing possibilities including lattice realizations of quantum Hall phenomena and non-abelian anyons at room temperature. However, these systems pose a number of important new theoretical challenges that arise due to the combined effects of interactions, band topology and the underlying lattice. Understanding this interplay, most crisply manifested through a varying Berry curvature, is a major objective of this proposal. To investigate these systems, we propose to use entanglement-based simulation approaches and diagnostics alongside with more traditional condensed matter theory approaches including series expansions, exact diagonalization, and band structure considerations.