Crystalline-equivalent topological phases of many-body fermionic systems in one dimension

We explore one-dimensional fermionic symmetry-protected topological (SPT) phases related by the crystalline equivalence principle. In particular, we study charge-conserving many-body topological phases of fermions protected respectively by chiral and reflection symmetries. While the classifications of the two crystalline-equivalent SPT phases are identical, their topological properties and phase structures can be very different, depending on the microscopic details. Specifically, we consider certain extensions of the Su-Schrieffer-Heeger model, with and without interactions, that preserve both chiral and reflection symmetries, and explicitly compute the many-body topological invariants based on the systems' ground states. The phase structures determined by these topological invariants align perfectly with the many-body spectra of deformations among the models. As expected, gapped deformations exist only when all the topological invariants remain unchanged. Moreover, we show that decomposable systems -- those that can be decomposed into local and decoupled subsystems -- can be topologically characterized by real-space quantum numbers directly associated with the symmetries. For reflection-symmetric systems, these quantum numbers are related to the many-body topological invariants via a bulk-center correspondence, which can be justified using the Atiyah-Hirzebruch spectral sequence in generalized homology theory. Finally, we discuss the role of transition symmetry in the many-body topologies of these SPT phases.