Classification of Average Crystalline Topological Superconductors through a Generalized Real-Space Construction

Abstract: We investigate a novel class of topological superconducting phases protected by exact fermion-parity symmetry and average crystalline symmetries. These phases belong to the broader class of average crystalline symmetry-protected topological (ACSPT) states and include numerous examples of intrinsic ACSPTs -- topological phases that arise only in the presence of disorder or decoherence. Unlike conventional symmetry-protected topological (SPT) phases, which require exact symmetry protection, average SPT (ASPT) phases remain robust as long as the symmetry is restored on average across disorder realizations or mixed-state ensembles. To classify these phases, we extend the real-space block state construction framework to account for average crystalline symmetries. In this generalized setting, lower-dimensional cells are decorated with ASPT phases, and the obstruction-free conditions are reformulated to incorporate the constraints imposed by average symmetry at block intersections. This provides a physically transparent and systematic method for classifying ASPTs with spatial symmetries that are only preserved statistically. We further validate our classification using a generalized spectral sequence analysis, which serves as an independent consistency check. Our results demonstrate that many crystalline topological superconductors remain well defined under realistic imperfections, and they uncover a rich landscape of intrinsically average-symmetry-protected phases that have no analog in clean systems.