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