Chirality polarizations and proof of spectral bulk-boundary correspondence

Surface physics determined by bulk properties is attracting attention these days. An important context is the electric polarization, where the surface charge is the bulk property modulo a quantum. A difference from the topological bulk-boundary correspondence (BBC) is that surface chage can take continuos values. Such a correspondence of continuous quantities in the bulk and the surface has not been fully explored.
It is proposed by numerics that surface accumulation of a component of Green’s function proportional to the chiral operator is a bulk property[1]. This relation is named as spectral BBC (SBBC). In the context of superconductivity, SBBC amounts to an interesting statement that surface odd-frequency Cooper pairs is a bulk property. However, formal proof has been lacking.
In this work, we prove SBBC via Wannier functions and the continuity equation of chirality charge. SBBC is explained as a electric-polarization-like BBC of chirality charges.
[1] S. Tamura, S. Hoshino, and Y. Tanaka, Phys. Rev. B 99, 184512 (2019)