Delicate topology of Luttinger semimetal

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 (LSM) is found to be a quantum critical point leading to a quantized jump of an integer-valued delicate topological invariant, i.e., the returning Thouless pump (RTP) index. 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. We further unveil that RTP physics widely exists in a large class of LSM-based Weyl materials that respect a rotoinversion symmetry, and we discuss their unconventional surface physics. Our work provides a starting point for exploring delicate topological phenomena in quantum materials.