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We study ultra-cold bosonic systems in optical lattice using quantum rotor approach to calculate the currentcurrent correlations, which provides the information about conductivity of the system. The method allows us to go beyond mean-field approximation and track the behavior of the real part of the conductivity along the phase transition between the Mott insulator and superfluid state for various geometries of the lattice. In the phaseordered state at zero temperature, a discrete ingredient appears resulting from the long-range coherence in the system.

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We calculate the conductivity of strongly interacting bosons described by the Bose-Hubbard model in a twodimensional lattice in controlled gauge potential acting as a uniform external magnetic field. In such scenario, two types of conductivity can be distinguished: intra-and inter-band. The interband contribution, usually omitted in analysis of multiband systems, appears to have a crucial role in the transport properties as its values are a few orders of magnitude greater than the intraband one.