Topologically distinct atomic insulators

Topological classification of quantum solids often (if not always) groups all trivial atomic or normal insulators (NIs) into the same featureless family. As we argue here, this is not necessarily the case always. In particular, when the global phase diagram of electronic crystals harbors topological insulators with the band inversion at various time-reversal invariant momenta ${\mathbf{K}}_{\mathrm{i}\mathrm{n}\mathrm{v}}^{\mathrm{T}\mathrm{I}}$ in the Brillouin zone, their proximal NIs display noninverted band-gap minima at ${\mathbf{K}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{N}\mathrm{I}}={\mathbf{K}}_{\mathrm{i}\mathrm{n}\mathrm{v}}^{\mathrm{T}\mathrm{I}}$. In such systems, once topological superconductors nucleate from NIs, the inversion of the Bogoliubov de Gennes bands takes place at ${\mathbf{K}}_{\mathrm{i}\mathrm{n}\mathrm{v}}^{\mathrm{B}\mathrm{d}\mathrm{G}}={\mathbf{K}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{N}\mathrm{I}}$, inheriting from the parent state. We showcase this (possibly general) proposal for two-dimensional time-reversal symmetry-breaking insulators. Then distinct quantized thermal Hall conductivity and responses to dislocation lattice defects inside the paired states (tied with ${\mathbf{K}}_{\mathrm{i}\mathrm{n}\mathrm{v}}^{\mathrm{B}\mathrm{d}\mathrm{G}}$ or ${\mathbf{K}}_{\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{N}\mathrm{I}}$), in turn unambiguously identify different parent atomic NIs.