# Topological state evolution by symmetry-breaking

## TL;DR

This paper develops a systematic framework to analyze how topological states in magnetic and non-magnetic materials evolve under symmetry-breaking, using a tree-like graph representation, supported by high-throughput data on magnetic materials.

## Contribution

It introduces a novel tree-like graph framework for topological state evolution under symmetry-breaking across all magnetic space groups, and applies it to a large dataset of magnetic materials.

## Key findings

- Different symmetry-breaking pathways lead to various contractions of the topological state graph.
- High-throughput analysis reveals a hierarchy of topological states along symmetry-breaking paths.
- Framework aids in guiding experimental tuning of band topology in materials.

## Abstract

Previous symmetry-based database searches have already revealed ubiquitous band topology in nature, while the destiny of band topology under symmetry-breaking is yet to be studied comprehensively. Here we first develop a framework allowing systematically ascertaining topological state evolution as expressed via a tree-like graph for magnetic/non-magnetic crystalline material belonging to any of the 1651 magnetic space groups. Interestingly, we find that specifying different ways of realizing symmetry-breaking leads to various contractions of the tree-like graph, as a new angle of comprehensively characterizing the correlation between a spontaneous symmetry-breaking and any symmetry-group-indicated physics consequence. We also perform a high-throughput investigation on the 1267 stoichiometric magnetic materials ever-experimentally synthesized to reveal a hierarchy of topological states along all continuous paths of symmetry-breaking (preserving the translation symmetry) from the parent magnetic space group to P1. The results in this work are expected to aid experimentalists in selecting feasible and appropriate means to tune band topology towards realistic applications and promote further studies on using tree-like graph to explore the interconnection between topology and other intriguing orderings.