Explicit equivalence between the spectral localizer and local Chern and winding markers

Abstract: Topological band insulators are classified using momentum-space topological invariants, such as Chern or winding numbers, when they feature translational symmetry. The lack of translation symmetry in disordered, quasicrystalline, or amorphous topological systems has motivated alternative, real-space definitions of topological invariants, including the local Chern marker and the spectral localizer invariant. However, the equivalence between these invariants is so far implicit. Here, we explicitly demonstrate their equivalence from a systematic perturbative expansion in powers of the spectral localizer's parameter $\kappa$. By leveraging only the Clifford algebra of the spectral localizer, we prove that Chern and winding markers emerge as leading-order terms in the expansion. It bypasses abstract topological machinery, offering a simple approach accessible to a broader physics audience.