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.

Figure 1

A flat band $\alpha$ with dispersion ${\epsilon }_{\alpha }$ (red) in the $\text{FBZ}$ experiences a LL spread upon applying a weak magnetic field $\delta B$ ending up with dispersion ${\tilde{\epsilon }}_{\alpha }$ (green). Even if the flat band ${\epsilon }_{\alpha }$ does not come in contact with a chemical potential $\mu$ (violet), its LL spread may dip or reach into it, rendering the band metallic.

Figure 2

(left) The bands of the unperturbed system are depicted in blue. The lower band is flat, and thus the candidate for our perturbation theory. (right) Closeup of the unperturbed flat band $\alpha$ in blue and its perturbed counterpart in green. Both plots use flux $\varphi /{\varphi }_0=0.1$ and the lattice spacing is set to unity.

Figure 3

(left) The curvatures ${\mathrm{\Omega }}^{\alpha }$ (blue) and ${\tilde{\mathrm{\Omega }}}^{\alpha }$ (orange) are depicted. (right) A section of the metric component in $xx$ direction at fixed $k_y=\pi /2$ is presented. Both plots use flux $\varphi /{\varphi }_0=0.1$ and the lattice spacing is set to unity.

Figure 4

(left) The metric component $g_{xx}^{\alpha }$ (blue) and ${\tilde{g}}_{xx}^{\alpha }$ (orange) are depicted. (right) A section of the curvatures at fixed $k_y=\pi /2$ is presented. Both plots use flux $\varphi /{\varphi }_0=0.1$ and the lattice spacing is set to unity.

Figure 5

The left plot depicts the emergent curvature ${\mathcal{U}}_{xy}$ and the right plot shows ${\mathcal{V}}_{xy}$. No magnetic field is included.