Design and characterization of all two-dimensional fragile topological bands

TL;DR

This paper introduces an algorithmic approach using evolutionary strategies to design and analyze all two-dimensional fragile topological bands, enabling scalable and efficient creation of topological materials across various physical platforms.

Contribution

The work presents a novel optimization algorithm for designing fragile topological bands, automating the process and broadening the scope of possible topological material configurations.

Findings

Successfully characterized fragile topological bands using the algorithm

Demonstrated the method's ability to optimize spectral quality and topological features

Expanded the potential for designing topological materials in various physical systems

Abstract

Designing topological materials with specific topological indices is a complex inverse problem, traditionally tackled through manual, intuition-driven methods that are neither scalable nor efficient for exploring the vast space of possible material configurations. In this work, we develop an algorithm that leverages the covariance matrix adaptation evolution strategy to optimize the Fourier representation of the periodic functions shaping the designer material's characteristics. This includes mass profiles or dielectric tensors for phononic and photonic crystals, respectively, as much as synthetic potentials applicable to electronic and ultra-cold atomic systems. We demonstrate our methodology with a detailed characterization of a class of topological bands known as "fragile topological", showcasing the algorithm's capability to address both topological characteristics and spectral quality. This automation not only streamlines the design process but also significantly expands the potential for identifying and constructing high quality designer topological materials across the wide range of platforms, and is readily extendable to other setups, including higher-dimensional and non-linear systems.