Local Classification of Fragile Topology

Fragile topology represents a new class of topological phases of matter in which the nontrivial topological features of a set of bands can be lost upon the addition of trivial bands. This class of topology has attracted considerable attention due to its manifestation in small-angle twisted bilayer graphene, where it shows significant relevance to correlated flat-band physics and superconductivity. Intriguingly, conventional schemes for diagnosing material topology, such as methods relying on the construction of exponentially localized Wannier functions, exhibit shortcomings for fragile topology. This leads to difficulties in establishing bulk-boundary correspondence due to the absence of general invariants, highlighting the need for a deeper understanding of material topology. Here, we propose a local classification of fragile topological systems through the spectral localizer framework defined from a system’s Hamiltonian and position operators. Through this real-space scheme, we present a local invariant and a quantitative measure of topological protection based on graded C∗-algebras that distinguishes systems based on whether they can be continued to a trivial atomic limit. Its real-space foundation enables the characterization of material topology in systems with finite size, disorder, and the influence from the gapless environments. We demonstrate the practical implementation of local classification by analyzing examples including the C₂T-symmetric twisted bilayer graphene model and photonic crystals embedded in air, which suggest that fragile topology can persist even if a heterostructure’s bulk spectral gap closes.