In the realm of condensed matter physics, the fractional quantum Hall effect stands as a singular experimental manifestation of topological order, characterized by the presence of anyons—quasiparticles that bear fractional charge and exhibit exchange statistics diverging from conventional fermions and bosons. This phenomenon, observed over four decades ago, was still missing the direct observation of similar topological orders arising purely from band structure—without the application of strong magnetic fields. In 2023 within the span of a few months, several pioneering experiments have illuminated this once theoretical domain. Studies on twisted homobilayer MoTe2 and pentalayer rhombohedral graphene placed on hBN have finally unveiled the existence of fractional Chern insulators (FCIs), the zero-magnetic field analog of fractional quantum Hall states.

The journey to this point, preceded by over a decade of theoretical frameworks and predictions surrounding FCIs, yet the experimental revelations have proved to be richer and more surprising than expected. In this talk, we will present how the combination of ab-initio and quantum many-body calculations can help us capture the different features observed in experiments. We will discuss the potential future for this exciting booming field, including the possible observation of fractional topological insulators, a yet-never observed topological ordered phase preserving the time reversal symmetry.