“Ten-fold way” table tells us that different spatial dimensions and symmetries correspond to different topological classes. However, this table only contains time-reversal symmetry, particle-hole symmetry and chiral symmetries, but not a crystalline symmetry. Discoveries for topological insulators, topological crystalline insulators, and high-order topological insulators make people realize that crystalline symmetries can bring new topological states and the diversity of symmetry-protected topological states in materials. Thus classification of topological states for the variety of crystalline symmetries becomes a vital issue.
In my talk, I will introduce a series of theoretical tools, such as compatibility relation, symmetry-based indicator, and indicator formulas[7-9], which can help us to diagnose various topological states by considering time-reversal symmetry and crystalline symmetries. After that, I will show you how to use those theoretical tools to perform a high-throughput calculation.