Band flatness optimization through complex analysis

TL;DR

This paper introduces a complex analysis-based method to optimize band flatness in topological electron systems, balancing flatness with hopping constraints, and illustrating it with fractional Chern insulator models.

Contribution

It presents a novel approach using Rouche's theorem to explicitly optimize band flatness considering topological constraints in electron models.

Findings

Optimized band flatness achieved through complex analysis techniques.

Explicit construction for two-band fractional Chern insulator models.

Topological properties can hinder band flatness due to geometric obstructions.

Abstract

Narrow band electron systems are particularly likely to exhibit correlated many-body phases driven by interaction effects. Examples include magnetic materials, heavy fermion systems, and topological phases such as fractional quantum Hall states and their lattice-based cousins, the fractional Chern insulators (FCIs). Here we discuss the problem of designing models with optimal band flatness, subject to constraints on the range of electron hopping. In particular, we show how the imaginary gap, which serves as a proxy for band flatness, can be optimized by appealing to Rouche's theorem, a familiar result from complex analysis. This leads to an explicit construction which we illustrate through its application to two-band FCI models with nontrivial topology (i.e. nonzero Chern numbers). We show how the imaginary gap perspective leads to an elegant geometric picture of how topological properties can obstruct band flatness in systems with finite ranged hopping.