Recent Developments in Fractional Chern Insulators

Fractional Chern insulators (FCIs) are lattice generalizations of the conventional fractional quantum Hall effect (FQHE) in two-dimensional (2D) electron gases. They typically arise in a 2D lattice without time-reversal symmetry when a nearly flat Bloch band with nonzero Chern number is partially occupied by strongly interacting particles. Band topology and interactions endow FCIs exotic topological orders which are characterized by the precisely quantized Hall conductance, robust ground-state degeneracy on high-genus manifolds, and fractionalized quasiparticles. Since in principle FCIs can exist at zero magnetic field and be protected by a large energy gap, they provide a potentially experimentally more accessible avenue for observing and harnessing FQHE phenomena. Moreover, the interplay between FCIs and lattice-specific effects that do not exist in the conventional continuum FQHE poses new theoretical challenges. In this chapter, we provide a general introduction of the theoretical model and numerical simulation of FCIs, then pay special attention on the recent development of this field in moir\'e materials while also commenting on potential alternative implementations in cold atom systems. With a plethora of exciting theoretical and experimental progress, topological flat bands in moir\'e materials such as magic-angle twisted bilayer graphene on hexagonal boron nitride have indeed turned out to be a remarkably versatile platform for FCIs featuring an intriguing interplay between topology, geometry, and interactions.