Stability and noncentered PT symmetry of real topological phases

Real topological phases protected by the spacetime inversion (P T) symmetry are a current research focus. The basis is that the P T symmetry endows a real structure in momentum space, which leads to Z2 topological classifications in 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 P T 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.