Engineering geometrically flat Chern bands with Fubini-Study Kähler structure
Bruno Mera, Tomoki Ozawa
Abstract
We describe a systematic method to construct models of Chern insulators whose Berry curvature and the quantum volume form coincide and are flat over the Brillouin zone; such models are known to be suitable for hosting fractional Chern insulators. The bands of Chern insulator models where the Berry curvature and the quantum volume form coincide, and are nowhere vanishing, are known to induce the structure of a Kähler manifold in momentum space, and thus we are naturally led to define to be Chern bands satisfying such properties. We show how to construct a geometrically flat Kähler band with Chern number equal to minus the total number of bands in the system, using the idea of Kähler quantization and properties of Bergman kernel asymptotics. We show that, with our construction, the geometrical properties become flatter as the total number of bands in the system is increased; we also show the no-go theorem that it is not possible to construct geometrically perfectly flat Kähler bands with a finite number of bands. We give an explicit realization of this construction in terms of theta functions and numerically confirm how the constructed Kähler bands become geometrically flat as we increase the number of bands. We also show the effect of truncating hoppings at a finite length, which will generally result in deviation from a perfect Kähler band but does not seem to seriously affect the flatness of the geometrical properties.