Topological quantum spin Hall semimetals with light

We introduce a quantum spin Hall semimetal or Fermi liquid characterized with a quantized ${\mathbb{Z}}_{2}$ topological invariant, measurable through circularly polarized light. We propose its engineering through two topological metallic band structures in crystals on the honeycomb lattice, with spin-orbit interaction, realizable from the interplay of a charge or spin density wave substrate and Zeeman effects, in between a quantum spin Hall and a quantum anomalous Hall insulators. Also, we address time-reversal symmetry and geometrical properties of the Fermi surface through light.