Interplay of band geometry and topology in ideal Chern insulators in the presence of external electromagnetic fields

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 first show that in the absence of external perturbations, novel Berry connections appear in ideal Chern insulating phases. Second, 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 a measurable quantum-mechanical correction in the Hall response. Our findings can be applied, for instance, to rhombohedral trilayer graphene at low energies.