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Symmetry indicators for topological materials

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Many free-fermion topological phases can be diagnosed by analyzing a suitable collection of symmetry data. While the Fu-Kane parity criterion for topological insulators is an early example, the systematic generalization to cover all possible crystalline symmetries and their associated topological phases has only recently been achieved. Interestingly, a key step in the generalization is to recognize the importance of understanding the entanglement-free product states: once these trivial phases are completely mapped, the topological ones are naturally exposed as the “remainders.” In this talk, we will first introduce the theory of symmetry indicators for topological (crystalline) insulators and semimetals, and then discuss how it can be generalized to cover superconductors.