Hall Conductivity of Strongly Interacting Bosons in Optical Lattice

We study the Hall conductivity of ultra-cold bosonic atoms in optical lattice described by the Bose-Hubbard model. We use the quantum rotor approximation, which allows to describe lattices with non-zero Chern numbers. As examples of such systems we consider square lattice in a synthetic magnetic field as well as the Haldane model. We derive formula for the Hall conductivity, which strongly resembles the famous TKNN formula. We investigate the behavior of conductivity depending on temperature and model parameters. It appears that bosonic systems substantially differ from the fermionic ones, e.g. the presence of non-zero Hall conductivity is not directly related to non-zero value of the Chern number.