  Starting from the Hofstadter butterfly, we define lattice versions of Landau
levels as well as a continuum limit which ensures that they scale to continuum
Landau levels. By including a next-neighbor repulsive interaction and
projecting onto the lowest lattice Landau level, we show that incompressible
ground states exist at filling fractions, $\nu = 1/3, 2/5 $ and $3/7$. Already
for values of $l_0/a \sim 2$ where $l_0$ ($a$) is the magnetic length (lattice
constant), the lattice version of the $\nu = 1/3$ state reproduces with nearly
perfect accuracy the the continuum Laughlin state. The numerical data strongly
suggests that at odd filling fractions of the lowest lattice Landau level, the
lattice constant is an irrelevant length scale. We find a new relation between
the hierarchy of incompressible states and the self-similar structure of the
Hofstadter butterfly.

Condensed Matter