Peru d’Ornellas : A unified perspective on noninteracting topology

Despite our mature understanding around non-interacting topological states of matter, there is currently no single method that can systematically deliver simple efficient topological invariants for all crystalline and non-crystalline systems alike. Here we revisit the spectral localizer methodology, showing how it can be treated it as a dimensional reduction scheme where the localizer’s zero-dimensional topology encodes the higher-dimensional phase of a parent Hamiltonian. Its classification reduces every topological invariant to a matrix signature or the sign of a Pfaffian, both of which are simple to interpret and efficient to compute in real space. We show that this can be applied regardless of the dimensionality or symmetry class of the system under consideration, demonstrating how it applies to strong, weak and rotationally invariant crystalline phases that were previously beyond the grasp of the spectral localizer formalism, atomic limits that escape scattering invariants, phases that evade symmetry-based indicator methods, as well as crystalline phases that had no previously known invariant in momentum space or otherwise.