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Recent advances in delicate topology have expanded the classification oftopological bands, but its presence in solid-state materials remains elusive.Here we show that delicate topology naturally emerges in the Luttinger-Kohnmodel that describes many semiconductors and semimetals. In particular, theLuttinger semimetal is found to be a quantum critical point leading to aquantized jump of an integer-valued delicate topological invariant. Away fromthis criticality, we have identified new types of electronic insulators andsemimetals with intertwined stable and delicate topologies. They all carrygapless surface states that transform anomalously under rotation symmetry. Ourwork provides a starting point for exploring delicate topological phenomena inquantum materials.

Delicate Topology of Luttinger Semimetal