Recent Developments in Fractional Chern Insulators.

Fractional Chern insulators (FCIs), initially conceptualized as lattice counterparts of fractional quantum Hall (FQH) states in the absence of a magnetic field, stand as a captivating frontier in condensed matter physics, illustrating a complex interplay between robust electron-electron interactions, band topology, and quantum geometry. This presentation provides an overview of recent key developments in the realm of FCIs, synthesizing cutting-edge theoretical advancements and remarkable experimental findings.

After over a decade of theoretical exploration, engineered moiré superlattice materials have recently emerged as an experimental platform for FCI physics. Milestone experimental results include weak-field FQH states in magic angle twisted bilayer graphene on boron nitride, originating from lattice FCIs rather than continuum Landau level states. Even more spectacular recent observations have identified authentic FCIs exhibiting a (zero-field) fractional quantum anomalous Hall effect in both twisted transition metal dichalcogenides and multilayer graphene.

The primary focus of this presentation is to elucidate the distinctions between moiré Chern bands and Landau levels, arising due to symmetry-breaking competing states as a consequence of the underlying quantum geometry in the moiré lattice setting. This platform also opens intriguing avenues for numerous new phenomena beyond Landau level physics, including FCIs in bands with higher Chern numbers, high-temperature FCIs, fractional anomalous Hall crystals combining charge density wave (CDW) order with anyon excitations, non-Abelian genons accompanying topological lattice defects, spin singlet and multicomponent FCIs, as well as a plethora of competing ordered single and multi-band states. This offers an exciting prospect where fundamental theory evolves through direct interaction with experiments, with first principles calculations serving as a bridge linking these two realms.