One-dimensional $\mathbb{Z}$-classified topological crystalline insulator under space-time inversion symmetry

We explore a large family of one-dimensional (1D) topological crystalline insulators (TCIs) classified by $\mathbb{Z}$ invariants protected by space-time inversion symmetry. This finding stands in marked contrast to the conventional classification of 1D band topology protected by inversion symmetry and characterized by $\mathbb{Z}_2$-quantized polarization (Berry-Zak phase). Such kind of enriched topological phases relies on imposing restriction on tunneling forms. By considering the nontrivial relative polarization among sublattices (orbitals), we introduce the inversion winding number as a topological invariant for characterizing and categorizing band topology. The bulk-edge correspondence with regard to the inversion winding number is discussed. Leveraging real-space analysis, we discover disorder-induced topological Anderson insulators and propose to experimentally distinguish band topology through relative polarization of edge states or bulk states. Our comprehensive findings present a paradigmatic illustration for the ongoing investigation and classification of band topology in TCIs.