  Ideal Chern insulating phases arise in two-dimensional systems with broken
time-reversal symmetry. They are characterized by having nearly-flat bands, and
a uniform quantum geometry -- which combines the Berry curvature and quantum
metric -- and by being incompressible. In this work, we analyze the role of the
quantum geometry in ideal Chern insulators aiming to describe transport in
presence of external out-of-plane magnetic and electric fields. We firstly show
that in the absence of external perturbations, novel Berry connections appear
in ideal Chern insulating phases. Secondly, we provide a detailed analysis of
the deformation of the quantum geometry once weak out-of-plane magnetic fields
are switched on. The perturbed Berry curvature and quantum metric provide an
effective quantum geometry, which is analyzed in the insulating regime and
provides an application of our novel connections. The conditions under which
the Girvin-MacDonald-Platzman algebra is realized in this situation are
discussed. Furthermore, an investigation of electrical transport due to the new
effective quantum geometry is presented once an electric field is added.
Restricting to the case of two bands in the metallic regime the quantum metric
appears as measurable quantum mechanical correction in the Hall response. Our
findings can be applied, for instance, to rhombohedral trilayer graphene at low
energies.
