Topological Lattice Models with Constant Berry Curvature

TL;DR

This paper investigates the possibility of achieving constant Berry curvature in topological lattice models, revealing that only models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature, but this does not necessarily enhance fractional Chern insulator states.

Contribution

It demonstrates that constant Berry curvature cannot be realized in two-band models and that flatbands with constant Berry curvature are fundamentally incompatible with Landau level algebra.

Findings

Constant Berry curvature is achievable only in models with three or more degrees of freedom per unit cell.

Achieving constant Berry curvature does not necessarily improve fractional Chern insulator properties.

Flatbands with constant Berry curvature cannot realize Landau level density algebra.

Abstract

Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of bosonic fractional Chern insulator states. In fact, we show that an "ideal flatband" cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.