中文摘要：

拓扑平带模型属于著名Haldane模型的扩展版本,至少有一个能带具有非平庸的拓扑性质,即有非零的陈数（Chem number）,另外,该能带的带宽很窄,且与其他能带间有较大能隙.通过对拓扑平带上强关联相互作用的费米子和玻色子晶格体系的系统数值研究,发现了一类新奇的阿贝尔型和非阿贝尔型分数量子霍尔效应.新发现的分数量子霍尔效应不同于传统朗道能级上的连续型分数量子霍尔效应,无须外加强磁场,有较大特征能隙,可在较高温度下存在,无需单粒子朗道能级,不能用常规Laughlin波函数来描述.这些无外加磁场、无朗道能级的分数化现象,定义了一类新的分数拓扑相,也称为分数陈绝缘体,其中的分数量子霍尔效应也称为分数量子反常霍尔效应.该新领域在近期引起了国际凝聚态物理学界的研究热情与广泛关注.对笔者与合作者在该领域的系列研究工作进行了综述介绍,以期引起国内外同行的进一步研究兴趣.

英文摘要：

Topological flat bands belong to the extensions of the well-known Haldane model, within which there is at least one energy band with non-trivial topological property, i. e. , has a nonzero Chern number. This energy band has a very narrow band-width, and is also separated from the other bands with large gaps. In recent systematic numerical studies of strongly correlated fermions and bosons in lattice systems with topologi- cal flat bands, a new class of exotic Abelian and non-Abelian fractional quantum Hall effect has been found. This newly found fractional quantum Hall effect is very different from the continuum fractional quantum Hall effect in conventional Landau levels, which can happen in the absence of an external strong magnetic field, has large characteristic gaps, can exist at high temperature, without single-particle Landau levels, and can not be described by conventional Laughlin wave functions. This intriguing fractionalization effect, without Landau levels and without external magnetic field, defines a class of fractional topological phases, also known as frac- tional Chern insulators, and the related fractional quantum Hall effect is also called fractional quantum anoma- lous Hall effect. This new research area has stimulated a lot of research activities and has also been widely concerned. This review aims to introduce our recent works done with collaborators in the area, and it is hoped to attract further research interest from domestic and abroad colleagues.