  We investigate the fate of hardcore bosons in a Harper-Hofstadter model which
was experimentally realized by Aidelsburger et al. [Nature Physics 11 , 162
(2015)] at half filling of the lowest band. We discuss the stability of an
emergent fractional Chern insulator (FCI) state in a finite region of the phase
diagram that is separated from a superfluid state by a first-order transition
when tuning the band topology following the protocol used in the experiment.
Since crossing a first-order transition is unfavorable for adiabatically
preparing the FCI state, we extend the model to stabilize a featureless
insulating state. The transition between this phase and the topological state
proves to be continuous, providing a path in parameter space along which an FCI
state could be adiabatically prepared. To further corroborate this statement,
we perform time-dependent DMRG calculations which demonstrate that the FCI
state may indeed be reached by adiabatically tuning a simple product state.
