K\"ahler geometry and Chern insulators: Relations between topology and the quantum metric

TL;DR

This paper explores the relationship between Kähler geometry and Chern insulators, revealing how the quantum metric and Berry curvature relate to the geometric structures of parameter spaces and their minimal volumes tied to topological invariants.

Contribution

It establishes conditions under which parameter spaces of Chern insulators attain minimal quantum volume, linking Kähler geometry to topological stability and fractional Chern insulators.

Findings

Minimal quantum volume of parameter spaces equals π times the Chern number.

Conditions for minimal volume relate to Kähler structures inherited from quantum state space.

In two-band systems, the Berry curvature's sign determines the minimality of the Brillouin zone volume.

Abstract

We study Chern insulators from the point of view of K\"ahler 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 K\"ahler 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 $\pi |\mathcal{C}|$, where $\mathcal{C}$ 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 non-degenerate, is achieved when the latter is endowed with the structure of a K\"ahler 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 K\"ahler 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.