Real topological phases protected by the space-time inversion ($PT$) symmetry are a current research focus. The basis is that the $PT$ symmetry endows a real structure in momentum space, which leads to ${\mathbb{Z}}_2$ topological classifications in one and two dimensions ($1\mathrm{D}$ and $2\mathrm{D}$). Here, we provide solutions to two outstanding problems in the diagnosis of real topology. First, based on the stable equivalence in $K$ theory, we clarify that the 2D topological invariant remains well defined in the presence of nontrivial 1D invariant, and we develop a general numerical approach for its evaluation, which was hitherto unavailable. Second, under the unit-cell convention, noncentered $PT$ symmetries assume momentum dependence, which violates the presumption in previous methods for computing the topological invariants. We clarify the classifications for this case and formulate the invariants by introducing a twisted Wilson-loop operator for both $1\mathrm{D}$ and $2\mathrm{D}$. A simple model on a rectangular lattice is constructed to demonstrate our theory, which can be readily realized using artificial crystals.

• Received 23 May 2022
• Revised 6 December 2022
• Accepted 11 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.195116