Origin of band flatness and constraints of higher Chern numbers

TL;DR

This paper introduces a wave function-based framework to classify and understand flat Chern bands, revealing fundamental constraints on higher Chern numbers and proposing strategies to overcome these limitations for practical realization.

Contribution

A new formalism for classifying flat bands based on wave functions, and insights into the constraints and strategies for realizing higher Chern number flat bands.

Findings

Perfectly flat Chern bands require infinite-range models.

Most natural Chern bands are of C=1, with higher Chern numbers being more complex to realize.

Strategies are proposed to bypass constraints on higher Chern numbers.

Abstract

Flat bands provide a natural platform for emergent electronic states beyond Landau paradigm. Among those of particular importance are flat Chern bands, including bands of higher Chern numbers ($C$$>$$1$). We introduce a new framework for band flatness through wave functions, and classify the existing isolated flat bands in a "periodic table" according to tight binding features and wave function properties. Our flat band categorization encompasses seemingly different classes of flat bands ranging from atomic insulators to perfectly flat Chern bands and Landau Levels. The perfectly flat Chern bands satisfy Berry curvature condition $F_{xy} = \text{Tr} \, \mathcal G_{ij}$ which on the tight-binding level is fulfilled only for infinite-range models. Most of the natural Chern bands fall into category of $C=1$; the complexity of creating higher-$C$ flat bands is beyond the current technology. This is due to the breakdown of the microscopic stability for higher-$C$ flatness, seen atomistically e.g. in the increase of the hopping range bound as $\propto$$\sqrt{C} a$. Within our new formalism, we indicate strategies for bypassing higher-$C$ constraints and thus dramatically decreasing the implementation complexity.