Kähler geometry and Chern insulators  relations between topology and the quantum metric
Abstract
We study Chern insulators from the point of view of Kähler 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 Kähler 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 twistangle 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 Kähler manifold inherited from the one of the space of quantum states. If the quantum volume of the twistangle torus is minimal, then both parameter spaces have the structure of a Kähler manifold inherited from the space of quantum states. These conditions turn out to be related to the stability of fractional Chern insulators. For twoband systems, the volume of the Brillouin zone is naturally minimal provided the Berry curvature is everywhere nonnegative or nonpositive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeroes due to topological constraints.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 arXiv:
 arXiv:2103.11583
 Bibcode:
 2021arXiv210311583M
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Quantum Gases;
 Mathematical Physics
 EPrint:
 10 pages, no figure