  The diamond and zinc-blende semiconductors are well-known and have been
widely studied for decades. Yet, their electronic structure still surprises
with unexpected topological properties of the valence bands. In this joint
theoretical and experimental investigation we demonstrate for the benchmark
compounds InSb and GaAs that the electronic structure features topological
surface states below the Fermi energy. Our parity analysis shows that the
spin-orbit split-off band near the valence band maximum exhibits a strong
topologically non-trivial behavior characterized by the $\mathcal{Z}_2$
invariants $(1;000)$. The non-trivial character emerges instantaneously with
non-zero spin-orbit coupling, in contrast to the conventional topological phase
transition mechanism. \textit{Ab initio}-based tight-binding calculations
resolve topological surface states in the occupied electronic structure of InSb
and GaAs, further confirmed experimentally by soft X-ray angle-resolved
photoemission from both materials. Our findings are valid for all other
materials whose valence bands are adiabatically linked to those of InSb, i.e.,
many diamond and zinc-blende semiconductors, as well as other related
materials, such as half-Heusler compounds.
