Design and characterization of all two-dimensional fragile topological bands

TL;DR

This work addresses designing all two-dimensional fragile topological bands by combining an automated, symmetry-aware bundling framework with a scalable design pipeline. It uses a CMA-ES–driven optimization of Fourier-encoded periodic coefficients to shape mass/dielectric profiles and potentials across phononic, photonic, and electronic platforms, while systematically ensuring fragile topology via irreducible representations and Chern-number constraints. A key contribution is the comprehensive, symmetry-driven bundling strategy across all eleven wallpaper groups hosting fragile roots, detailing how mirrors, glides, and TRS interact with Chern parity to form robust band bundles and avoid unwanted gap closings. The framework enables automated discovery of high-quality designer topological materials across multiple physical platforms and lays groundwork for extending to higher dimensions and nonlinear systems, with explicit examples for phonons, TM/TE photons, and Schrödinger electrons. These advances offer a path toward scalable, programmable realization of fragile topological bands in practical devices.

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.

Figures (43)

• Figure S1: Left: Unit cell with the full symmetry of $p2$. Middle: General positions of $p2$. Right: Brillouin zone with the high-symmetry points and lines indicated.
• Figure S2: One sample for each of the roots in $p2$ for phonons and TM photons.
• Figure S3: One sample for each of the roots in $p2$ for systems described by the Schrödinger equation.
• Figure S4: One sample for each of the roots in $p2$ for TE photons.
• Figure S5: Left: Unit cell with the full symmetry of $p2mm$. Middle: General positions of $p2mm$. Right: Brillouin zone with the high-symmetry points and lines indicated.