Hamiltonian Theory of Fractional Chern Bands

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 It has been known for some time that
a system with a filled band will have an integer quantum Hall conductance equal
to its Chern number, a toplogical index associated with the band. While this is
true for a system in a magnetic field with filled Landau Levels, even a system
in zero external field can exhibit the QHE if its band has a Chern number. I
review this issue and discuss a more recent question of whether a partially
filled Chern band can exhibit the Fractional QHE. I describe the work done with
Ganpathy Murthy in which we show how composite fermions, which were so useful
in explaining the usual FQHE, can be introduced here and with equal success by
adapting our Hamiltonian Theory of CFs developed for the FQHE in the continuum.