Kahler geometry and Chern insulators: Relations between topology and the quantum metric

We study Chern insulators from the point of view of Kahler geometry, i.e., the geometry of smooth manifolds equipped with a compatible triple consisting of a symplectic form, an integrable almost complex structure, and a Riemannian metric. The Fermi projector, i.e., the projector onto the occupied bands, provides a map to a Kahler manifold. The quantum metric and Berry curvature of the occupied bands are then related to the Riemannian metric and symplectic form, respectively, on the target space of quantum states. We find that the minimal volume of a parameter space with respect to the quantum metric is pi vertical bar C vertical bar, where C is the first Chern number. We determine the conditions under which the minimal volume is achieved both for the Brillouin zone and the twist-angle space. The minimal volume of the Brillouin zone, provided the quantum metric is everywhere nondegenerate, is achieved when the latter is endowed with the structure of a Kahler manifold inherited from the one of the space of quantum states. If the quantum volume of the twist-angle torus is minimal, then both parameter spaces have the structure of a Kahler manifold inherited from the space of quantum states. These conditions turn out to be related to the stability of fractional Chern insulators. For two-band systems, the volume of the Brillouin zone is naturally minimal provided the Berry curvature is everywhere non-negative or nonpositive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeros due to topological constraints.

Automated summary

In this study of Chern insulators, the relationship between quantum metric, Berry curvature, Riemannian metric, and symplectic form is explored, showing that the minimal volume of parameter space is related to the first Chern number. The conditions for achieving minimal volume in both Brillouin zone and twist-angle space are also determined, with implications for the stability of fractional Chern insulators. Additionally, it is found that for two-band systems, the volume of the Brillouin zone is minimal under certain conditions related to Berry curvature and topological constraints.