Robust Signatures of Fragile Topology (opens in new tab)

Some topological phases of matter are called fragile. They do not generically host protected gapless boundary states and they can be trivialized by adding additional valence bands. Here we show that fragile topology nevertheless has robust signatures: it yields bulk Dirac cones in two-dimensional materials with arbitrarily many bands. In systems with time-reversal and twofold rotation symmetry, we establish a fragile topological index which guar...