Stable real-space invariants and topology beyond symmetry indicators

We introduce stable real-space invariants (SRSIs), topological invariants defined from adiabatic deformations between Wannier states, generalizing previously discovered local and composite real-space invariants. SRSIs are $Z$ - and $Z_n$ -valued (n = 2, 4) linear combinations of Wannier state multiplicities characterizing the stable equivalence of atomic insulators. We enumerate all SRSIs in nonmagnetic space groups with and without spin-orbit coupling. $Z$ SRSIs are in one-to-one correspondence with momentum-space symmetry data and thus determine symmetry indicators of topology (SIs). $Z_n$ SRSIs capture real-space information beyond momentum-space symmetry data and SIs. Applying SRSIs to split elementary band representations (EBRs) whose symmetry data decomposes into positive sums of other EBR symmetry data, we diagnose the topology of all 211 cases across 51 space groups except for 8 exceptions in 5 space groups. Our results solidify Topological Quantum Chemistry beyond SIs and momentum-space symmetry data. Finally, we use SRSIs to diagnose an obstructed atomic insulator in a realistic material.