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

在广泛的凝聚态系统中，包括强磁场中的二维超冷电子(量子霍尔系统)，以及具有竞争相互作用的受挫磁性材料，预测了在低能尺度上出现带有分数量子数的新型量子有序和激发。这些场之间快速发展的界面，特别是自旋-轨道耦合材料中相互作用的拓扑相，是这一提议的重点。除了具有根本意义外，这些系统还可能掌握未来技术发展的关键，如拓扑量子计算。这些系统中的一个中心挑战是在量子力学层面上理解相互作用和拓扑之间的微妙相互作用。在受挫的量子磁体中，拓扑通过非平凡的晶格结构表现出来，导致许多近乎简并态，以及由此产生的丰富而复杂的低能行为。正如最近才充分认识到的那样，自旋-轨道耦合为受挫的系统增加了新的方面，特别是包括虹彩星座。在最近和正在进行的有趣实验活动的推动下，这一提议旨在更好地理解价键晶体和自旋液体之间的竞争，并提供这些场景的可观察到的签名。该拓扑也可以表现为非平凡带结构的形式，就像在拓扑绝缘体的情况下一样。除了早期的量子霍尔系统，在量子霍尔系统中，复杂的多体态是由于巨大简并的朗道能级内的残余相互作用而形成的，到目前为止，焦点一直集中在基本上非相互作用的实现上。然而，一类新的高关联态-分数拓扑绝缘体-最近被认为是在具有非平凡拓扑的几乎无色散带的晶格系统中形成的。这种带可以在自旋-轨道耦合的固态材料中实现，并且不需要外部磁场。原则上，这开启了许多有趣的可能性，包括室温下量子霍尔现象和非阿贝尔任意子的晶格实现。然而，由于相互作用、能带拓扑和基本晶格的综合影响，这些系统提出了一些重要的新的理论挑战。理解这种相互作用，最清楚地表现在不同的浆果曲率，是这项提议的主要目标。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.