Brillouin zone folding method for quasiperiodic superconductivity in multilayer systems: application to electronic structure and optical responses

We construct an efficient momentum space approach to the superconductivity in quasiperiodic multilayer systems. To this end, we extend the Brillouin zone (BZ) folding method to the superconducting (SC) phases by formulating the gap equation in the momentum space representation with the BZ folding. We show that the physical observables in quasiperiodic multilayers are generally given by so-called quasiperiodic functions. Consequently, there appear SC order parameters with finite momenta in the BZ folding method, which corresponds to the spatial fluctuation of the SC order in quasiperiodic multilayers. We also find a systematic way to compute the physical observables in the BZ folding method with a proper normalization condition using the continued fraction approximation. We apply the BZ folding method to a one-dimensional toy model of quasiperiodic superconductors and demonstrate its numerical efficiency compared to the conventional method based on the real space representation with a large system size. We also study a quasiperiodic bilayer system of Rice-Mele model and an s-wave superconductor to demonstrate an efficient computation of optical responses of a quasiperiodic superconductor with inversion symmetry breaking.