Engineering geometrically flat Chern bands with Fubini-Studyler structure

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 Kahler manifold in momentum space, and thus we are naturally led to define Ktihler bands to be Chern bands satisfying such properties. We show how to construct a geometrically flat Kahler band with Chern number equal to minus the total number of bands in the system, using the idea of Kahler 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 Kahler 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 Kahler 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 Kahler band but does not seem to seriously affect the flatness of the geometrical properties.

Automated summary

We describe a systematic method to construct models of Chern insulators with Berry curvature and the quantum volume form coinciding and being flat, which can host fractional Chern insulators. The construction of geometrically flat Kahler bands with Chern number equal to minus the total number of bands is shown, but it is not possible to construct geometrically perfectly flat Kahler bands with a finite number of bands. The effect of truncating hoppings at a finite length is also explored, showing some deviation from perfect Kahler bands but no significant impact on the flatness of geometrical properties.