Stable Real-Space Invariants: Topological Quantum Chemistry Beyond Symmetry Indicators

Band topology is characterized by properties invariant under smooth deformations that respect all symmetries of a given space group. Under this definition, there even exist distinct atomic insulators that cannot be smoothly deformed into each other. Momentum space topological invariants like symmetry indicators (SIs) do not distinguish between all topologically distinct sets of bands and fail to identify large classes of "non-symmetry-indicated" topological phases. To address these limitations, in this talk we introduce stable real-space invariants (SRSIs), which incorporate global position space connectivity constraints to enhance the previously proposed local real-space invariants. SRSIs are linear combinations of Wannier orbital multiplicities and are Z- and Zn-valued invariants. We show that SRSIs capture stable topological equivalence between bands, meaning that two insulators with matching SRSIs are smoothly deformable into each other upon the inclusion of auxiliary trivial orbitals. We will discuss how this framework extends topological quantum chemistry beyond SIs and has many applications. We argue that the SRSIs encode the existing theory of SIs. Lastly, we show how Zn-valued SRSIs can detect topologies even when the SIs of bands above and below band gap are trivial.