Chirality polarizations and spectral bulk-boundary correspondence

Surface physics dominated by bulk properties has been one of the central interests in modern condensed matter physics, from electric polarization to bulk-boundary correspondence of topological insulators and superconductors. Here we extend theory of electric polarization to chirality polarizations, that is, surface charges corresponding to local antisymmetries characterized as a bulk property. Using the notion of chirality polarizations, we prove the recently proposed spectral bulk-boundary correspondence, a generalization of bulk-boundary correspondence in one-dimensional chiral symmetric systems into complex frequencies. We show a physically transparent proof via Wannier functions and a formal proof by considering the adiabatic change of surface chirality charges, highlighting the similarities and the differences between electric polarization and chirality polarizations.