Topological phase engineering of neutral bosons loaded in an optical lattice opens a new window for manipulating of transport phenomena in such systems. Exploiting the Bose Hubbard model and using the magnetic Kubo formula proposed in this paper we show that the optical conductivity abruptly changes for different flux densities in the Mott phase. Especially, when the frequency of the applied field corresponds to the on-site boson interaction energy, we observe insulator or metallic behavior for a given Hofstadter spectrum. We also prove, that for different synthetic magnetic field configurations, the critical conductivity at the tip of the lobe is non-universal and depends on the energy minima of the spectrum. In the case of 1/2 and 1/3 flux per plaquette, our results are in good agreement with those of the previous Monte Carlo (MC) study. Moreover, we show that for half magnetic-flux through the cell the critical conductivity suddenly changes in the presence of a superlattice potential with uniaxial periodicity.

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An even number of fermions can behave in a bosonic way. The simplest scenario involves two fermions which can form a single boson. But four fermions can either behave as two bipartite bosons or further assemble into a single four-partite bosonic molecule. In general, for 2N fermions there are many possible arrangements into composite bosons. The question is: what determines which fermionic arrangement is going to be realized in a given situation and can such arrangement be considered truly bosonic? This work aims to find the answer to the above question. We propose an entanglement-based method to assess bosonic quality of fermionic arrangements and apply it to study how the ground state of the extended one-dimensional Hubbard model changes as the strength of intra-particle interactions increases.