A Spatial Localizer for electrons in insulators

The representation of insulators in terms of their electronic Wannier functions has been central to understanding polarization, orbital magnetization, and topological classification. Yet, it breaks down in phases where exponentially localized Wannier states cannot be defined. In this talk, we introduce the Spatial Localizer—an operator that embeds multidimensional generalizations of the Resta position operator within Clifford algebras—to construct both Wannier and non-Wannier local state representations across all topological phases. This framework provides a gauge-invariant and symmetry-agnostic route to obtain maximally localized states within the occupied Hilbert subspace under both open and periodic boundary conditions. These properties suggest the Spatial Localizer may be useful for studying disorder, real-space topology, and the onset of strongly interacting topological phases. We discuss its formal relation to spread-minimization methods and the quantum metric, and illustrate its application to several topological insulators.