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First order Quantum Hall Transitions in Hofstadter Butterfly in the Honeycomb Lattice

We analyze the effects of nearest neighbor repulsive interactions in the Hofstadter system in a honeycomb lattice. At low fillings, we show that, as the interaction strength is increased there are two first order transitions, a Landau transition with translational and rotational symmetries broken, followed by a topological transition with a jump in the quantized Hall conductivity. We therefore predict that in physical realizations where the interaction effects are strong, there would be translation symmetry broken states with quantized Hall conductivities that differ from those predicted by the non-interacting theory.