We show via tensor network methods that the Harper-Hofstadter Hamiltonian forhard-core bosons on a square geometry supports a topological phase realizingthe $\nu=1/2$ fractional quantum Hall effect on the lattice. We address therobustness of the ground state degeneracy and of the energy gap, measure themany-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 thetopological entanglement entropy by taking a combination of latticebipartitions that reproduces the topological structure of the originalproposals by Kitaev and Preskill, and Levin and Wen. The numerical results showthat the topological contribution is compatible with the expected value $\gamma= 1/2$. Our results provide extensive evidence that FQH states are within reachof state-of-the-art cold atom experiments.
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