Topological phases protected by projective space-time inversion symmetry in alkaline-earth-metal-like atoms

An important aspect in categorizing topological phases is whether the system is spinless or spinful, given that these classes exhibit distinct symmetry algebras, leading to disparate topological classifications. By utilizing the projective presentation strategy, the topological phases of spinless (or spinful) systems can be emulated using spinful (or spinless) systems augmented with gauge fields. In this study, we propose to implement the topological phases safeguarded by the unique projective space-time inversion symmetry inherent to spinful models, using synthetic spinless alkaline-earth-metal-like atoms. Employing the separation of orbital and nuclear-spin degrees of freedom, the model is configured as a rectangular tube penetrated by a uniform magnetic flux through each plaquette, which simulates a spinless ladder endowed with projective space-time inversion symmetry satisfying the algebraic properties of a spinful model. For interacting topological phases with interorbital spin-exchange interactions, which also adhere to space-time inversion symmetry, the four-fold degeneracy of edge modes is split into two pairs of edge modes with two-fold degeneracy. We map the complete phase diagram in the end and discover that these interacting topological phases ultimately evolve into distinct charge-density-wave phases via spontaneous symmetry breaking.