  We show via tensor network methods that the Harper-Hofstadter Hamiltonian for
hard-core bosons on a square geometry supports a topological phase realizing
the $\nu=1/2$ fractional quantum Hall effect on the lattice. We address the
robustness of the ground state degeneracy and of the energy gap, measure the
many-body Chern number, and characterize the system using Green functions,
showing that they decay algebraically at the edges of open geometries,
indicating the presence of gapless edge modes. Moreover, we estimate the
topological entanglement entropy by taking a combination of lattice
bipartitions that reproduces the topological structure of the original
proposals by Kitaev and Preskill, and Levin and Wen. The numerical results show
that the topological contribution is compatible with the expected value $\gamma
= 1/2$. Our results provide extensive evidence that FQH states are within reach
of state-of-the-art cold atom experiments.
