The Harper-Hofstadter model provides a fractal spectrum containing topological bands of any integer Chern number $C$. We study the many-body physics that is realized by interacting particles occupying Harper-Hofstadter bands with $|C|>1$. We formulate the predictions of Chern-Simons or composite fermion theory in terms of the filling factor $\nu$, defined as the ratio of particle density to the number of single-particle states per unit area. We show that this theory predicts a series of fractional quantum Hall states with filling factors $\nu =r/(r|C|+1)$ for bosons, or $\nu =r/(2r|C|+1)$ for fermions. This series includes a bosonic integer quantum Hall state in $|C|=2$ bands. We construct specific cases where a single band of the Harper-Hofstadter model is occupied. For these cases, we provide numerical evidence that several states in this series are realized as incompressible quantum liquids for bosons with contact interactions.

• Received 5 May 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.126401