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.

Figure 1

(a) A schematic figure of electric charge density of the system under considerations. We show $\overline{\rho }\left(x\right)$ only around $x=L$. Similar figures are obtained for chirality charges by replacing $\rho \left(x\right)$ with ${\rho }_f(x,z)$. (b) Intuitive picture for the origin of microscopic excess charge bound at the left end [33]. In (a) and (b) we ignored surface modification of GWFs from the Wannier functions in PBC, for simplicity.

Figure 2

(a)–(c) An antiferromagnetic chain with $L=5a$ and several possible sets of its GWFs. (b) GWFs are assigned to each lattice point in the same way as (a). GWFs with positive spin expectation values are highlighted by red and pink, while those with negative spin expectation values are highlighted by blue and light blue. GWFs colored by gray have a vanishing spin expectation value.