The Hall response provides an important characterization of strongly correlated phases of matter. We study the Hall conductivity of interacting bosons on a lattice subjected to a magnetic field. We show that for any density or interaction strength, the Hall conductivity is characterized by an integer. We find that the phase diagram is intersected by topological transitions between different values of this integer. These transitions lead to surprising effects, including sign reversal of the Hall conductivity and extensive regions in the phase diagram where it acquires a negative sign, which implies that flux flow is reversed in these regions—vortices there flow upstream. Our findings have immediate applications to a wide range of phenomena in condensed matter physics, which are effectively described in terms of lattice bosons.
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