Figure 1:
Contrast between the “Landau level” mechanism of realizing FQH states (left column) and the “Chern insulator” mechanism (right column). (a) and (b) show examples of band structures. The traditional mechanism involves Landau levels, each with a Hall conductivity (given by the Chern number C ) of the same sign (a). In Chern-band systems, the bands have different Chern numbers (b). In real space, an effective magnetic flux threads each elementary plaquette (i.e., each little square). In the lattice version of Landau levels, each flux has the same sign (c), just as the effective magnetic field in a trapped rotating Bose gas (e) has the same sign everywhere. In Chern insulators, fluxes often alternate (d), as through a scheme like the one proposed in Ref.[2] using polar molecules in optical lattices (f).