Stability and noncentered $PT$ symmetry of real topological phases

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 $Z_2$ topological classifications in one and two dimensions ($1D$ and $2D$). 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 $1D$ and $2D$. A simple model on a rectangular lattice is constructed to demonstrate our theory, which can be readily realized using artificial crystals.