Simple invariants for band topology

Despite the exhaustive understanding gathered around non-interacting topological states of matter, there is no single method capable of systematically delivering simple, numerically efficient topological invariants that is applicable to all crystalline and non-crystalline systems alike. Here we revisit the spectral localizer operator, constructed from the Hamiltonian and position operators, and show how it can be treated it as an auxiliary zero-dimensional Hamiltonian whose topology encodes the higher dimensional phases of the parent Hamiltonian. Its classification reduces every topological invariant to a matrix signature or the sign of a Pfaffian for an appropriate localizer, both of which are simple to interpret and efficient to compute in real space. We validate this approach by deriving simple real-space invariants for 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 phases that had no previously known invariant. Our work provides a systematic way to construct any non-interacting topological invariant for a crystalline or non-crystalline systems, opening avenues to classify and predict the topology of previously unexplored classes of materials.