Chirality polarization and spectral bulk-boundary correspondence

Surface physics dominated by bulk properties is attracting much attention. For example, bulk-boundary correspondence predicts topological boundary states. Another important context is the electric polarization: Surface charge is the bulk property modulo a quantum [1]. This relation is analogous to BBC, in the sense that physics apparently sensitive to the surface details is the bulk property. It is an interesing question whether this "BBC" can be generalized to the other physical quantities.
Recently, Tamura et al. proposed by numerics that surface accumulation of odd-frequency Cooper pairs is the bulk property [2]. This relation is named spectral bulk-boundary correspondence (SBBC). However, formal proof for SBBC has been lacking.
In this work, we prove SBBC based on the analogy with electric polarization: SBBC can be regarded as polarization of chiralities.

[1] D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442 (1993).
[2] S. Tamura, S. Hoshino, and Y. Tanaka, Phys. Rev. B 99, 184512 (2019).