Robust Signatures of Fragile Topology in Nonmagnetic Materials

Some topological phases of matter are called fragile. They can be trivialized by adding additional valence bands and do not generically host protected gapless boundary states. Here we establish a fragile topological index with a bulk signature: it guarantees the presence of gap closing points in the bulk band structure of systems with both time-reversal and twofold rotation symmetry. This result extends the established use of the Euler class in diagnosing the presence of Dirac cones for general two-dimensional systems with a combination of time-reversal and twofold rotation symmetry, to the case of nonmagnetic materials obeying both symmetries individually. The abundance of natural nonmagnetic materials with these symmetries renders the topological invariant practically useful, despite its fragility. We demonstrate this by applying first-principles calculations to five materials and finding that all of them host Dirac points identified by the invariant, suggesting its widespread applicability. Our results thus provide a robust spectroscopic signature of fragile topology in nonmagnetic systems which can be directly accessed in experiments on practically relevant materials. Furthermore, they enable an alternate method for finding Dirac cones in ab-initio simulations, and may be used as a route towards identifying materials with nonlinear transport properties.