Strange Correlator as a Diagnostic of Fragile and Delicate Topology

The strange correlator provides a diagnostic for topological order by evaluating correlation functions between two states with potentially different topological characters. In free-fermion systems, momentum-space divergences correspond to singularities in the overlap between occupied Bloch subspaces, reflecting topological obstructions between the states. We examine this behavior in models exhibiting fragile and delicate topology, where conventional stable invariants vanish. Consistent divergence patterns emerge across these systems, preventing smooth deformation between phases. These results suggest that the strange correlator can detect nontrivial band topology even in phases lacking stable indices, offering a unified framework for probing both stable and unstable topological phases.