Lost of Bosonic Coherence Due to the Fermionic Presence Under the Synthetic Magnetic Field

We have studied effects of inter-species interaction of the bosons in a Bose-Fermi mixture over a large regime of particle numbers under the synthetic magnetic field. Analytically derived formulas for the density of states for several values of the magnetic fields f=1/2,1/3,1/4,1/6,1/8 and 3/8 allows us to calculate, with a very good accuracy, the effective interaction between bosons. The presence of the Hofstadter butterfly spectrum and fermionic species induces alternating sign potential between neutral bosonic atoms. Consequently bosons can attract or repulse each other whether sign of it strongly depends on the strength of the magnetic field applied to the sample.