Position-space invariants for topological crystalline insulators

The study of topological phases of crystalline insulators leads to the problem of characterizing homotopy classes of Fermi projections associated with gapped Schrödinger operators with space-group symmetries. The Bloch–Floquet decomposition provides a momentum-space description of the occupied states, in which crystalline symmetries give rise to topological invariants that distinguish phases under deformations preserving the spectral gap and the symmetries. The usual formulation of these invariants relies on this momentum-space description and on compatibility of the Fermi projection with the prescribed space-group. This motivates the search for a position-space formulation whose behavior can also be studied when a momentum-space description is unavailable or the symmetries are broken.

In this talk, I will present an operator-theoretic framework for expressing crystalline symmetry invariants directly in position space. The approach combines propagation properties and local trace-class regularity of Fermi projections with Fourier-analytic methods to derive reconstruction formulas relating position-space quantities to the invariants of a crystalline system. I will also discuss the stability of these quantities under perturbations of the Hamiltonian that preserve the spectral gap, highlighting the role of the spatial distribution of the perturbation.