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

点击翻译按钮获取中文摘要

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.