A fractional Chern insulator (FCI) is a lattice counterpart of the fractional quantum Hall state. The ordinary fractional quantum Hall effect requires a two-dimensional electron gas in a strong real magnetic field, forming highly degenerate, dispersionless Landau levels that interactions then fill fractionally. An FCI reaches an analogous topologically ordered phase on a lattice instead: particles hop on a lattice threaded by an artificial (synthetic) magnetic flux per plaquette, producing a topologically nontrivial, sufficiently flat band (a Chern band) that interactions can fractionally fill in much the same way a Landau level does — without needing a literal continuum magnetic field or the low temperatures and high sample mobility a real fractional quantum Hall measurement demands.
Because the flux and interactions are both engineered rather than found in a fixed material, FCIs are a natural target for quantum-simulation platforms — ultracold atoms in optical lattices, photonic lattices, or moiré materials — where the flux per plaquette and interaction strength can be tuned directly. Like their continuum counterpart, FCIs host anyonic quasiparticle excitations (quasiholes and quasielectrons) that carry a fraction of the elementary charge and obey exchange statistics that interpolate between bosons and fermions; creating, localizing, and moving individual anyons under direct control is a central experimental goal, since it is a prerequisite for using their exotic statistics for topological quantum information processing.