# Fragile Phases As Affine Monoids: Classification and Material Examples

## TL;DR

This paper provides the first complete classification of fragile topological phases in electronic structures, linking their mathematical structure to affine monoids and identifying real materials exhibiting these phases.

## Contribution

It introduces a comprehensive classification of fragile topological phases via symmetry eigenvalues and connects this to affine monoids, also predicting real materials with such phases.

## Key findings

- Enumerated hundreds of thousands of fragile topological phases
- Linked fragile phases to affine monoids mathematically
- Predicted and identified real materials with fragile topological bands

## Abstract

Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an Elementary Band Representation (Atomic Limit Band) splits into a particular set of bands. We obtain, for the first time, a complete classification of the fragile topological phases which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure which far surpasses that of stable/strong topological states. We find and enumerate all hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues (available at http://www.cryst.ehu.es/html/doc/FragileRoots.pdf), and link the mathematical structure of these phases to that of Affine Monoids in mathematics. Furthermore, we predict and calculate, for the first time, (hundred of realistic) materials where fragile topological bands appear, and show-case the very best ones.