Where is an electron in an insulator? In a perfect infinite crystal, the answer involves Wannier functions — localized wavepackets constructed from Bloch states. But real insulators have edges, defects, and disorder. The Wannier construction breaks down at boundaries because there's no translational symmetry left to exploit.
Cerjan & Loring (arXiv:2603.13206) introduce a spatial localizer: a position-space operator that answers “where are the electrons?” without needing periodicity. The framework uses spectral properties of quantum-mechanical operators to assign electron positions even in systems with realistic boundaries and defects.
The striking result: the answer depends on what kind of insulator you're asking about. In ordinary atomic insulators, the localizer recovers maximally localized Wannier functions — the same answer the bulk theory gives, extended gracefully to the boundary. But in Chern insulators — topological materials where the bulk carries a quantized Hall conductance — the localizer produces something entirely different: coherent states analogous to those found in quantum Hall systems. Gaussian wavepackets, not exponentially localized orbitals.
The electron's “address” is determined by the topology of the material, not just its chemistry. In a trivial insulator, electrons live at sharp addresses — this atom, that bond. In a topological insulator, electrons at the boundary have fuzzy addresses that reflect the non-trivial winding of the bulk band structure. You can't localize what topology has delocalized.
The framework makes something conceptually clean that was previously argued by analogy: topology doesn't just classify bulk properties. It determines the spatial character of individual electronic states, all the way down to how sharply you can say “the electron is here.”