Fate of Majorana modes in ${\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{I}\mathrm{n}}_5$/${\mathrm{Y}\mathrm{b}\mathrm{C}\mathrm{o}\mathrm{I}\mathrm{n}}_5$ superlattices -a test bed for reduction of topological classification-

In this paper, we propose $\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{I}{\mathrm{n}}_5\mathrm{/}\mathrm{Y}\mathrm{b}\mathrm{C}\mathrm{o}\mathrm{I}{\mathrm{n}}_5$ superlattice systems as a test bed for the reduction of topological classification in free fermions. We find that the system with quad-layer of $\mathrm{C}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{I}{\mathrm{n}}_5$ shows a topological crystalline superconducting phase with the mirror Chern number eight at the non-interacting level. Furthermore, we demonstrate that in the presence of two-body interactions, gapless edge modes are no longer protected by the symmetry in the system with quad-layer, but are protected in the systems with bi- or tri-layer. This clearly exemplifies the reduction of topological classification from $\mathbb{Z}\oplus \mathbb{Z}$ to $\mathbb{Z}\oplus {\mathbb{Z}}_8$.