Stable Real-Space Invariants and Topology Beyond Symmetry Indicators

We show that certain band gaps, which appear topologically trivial from the perspective of symmetry indicators (SIs), must instead be topological, as guaranteed by real-space information that follows from Topological Quantum Chemistry (TQC). To address this, we introduce stable real-space invariants (SRSIs) that generalize the previously discovered local and composite real-space invariants to global topological invariants of a given set of bands. These are linear combinations of Wannier state multiplicities at Wyckoff positions and take the form of $\mathbb{Z}$- and $\mathbb{Z}_n$-valued quantities ($n=2,4$). We enumerate all $\mathbb{Z}$SRSIs and $\mathbb{Z}_n$SRSIs in all non-magnetic space groups (SGs) with and without spin-orbit coupling. SRSIs fully diagnose the stable equivalence of atomic insulators, ensuring that two atomic insulators with matching SRSIs are adiabatically deformable to one another in the presence of auxiliary trivial bands. For both atomic and topological bands, $\mathbb{Z}$SRSIs are determined by the momentum-space symmetry data and thus determine the SIs. $\mathbb{Z}_n$SRSIs provide additional information about trivial band structures not captured by momentum-space data. While split elementary band representations (EBRs), where the bands forming an EBR split into disconnected parts, must induce band topology, there are 211 cases across 51 SGs where the momentum-space data of an EBR decomposes linearly with positive integer coefficients into those of other EBRs. We demonstrate that $\mathbb{Z}_n$SRSIs successfully identify the band topology in the majority of these split EBR cases, diagnosing all but 8 cases in 5 SGs. Our results solidify the conceptual framework of TQC as containing, but going beyond, SIs and momentum-space symmetry data.