  High-order harmonic generation (HHG) from atoms is a coherent light source
that opens up attosecond physics, but it is the application of HHG to solids
that brings much of excitement for the last decade. Here we report a completely
new kind of harmonics in a quasi-one-dimensional and hexagonal barium titanium
sulfide: Under circularly polarized laser excitation, harmonics are generated
only at first, fifth, seventh and eleventh orders. These magic harmonics appear
only with circularly polarized light, not with linearly polarized light.
Neither cubic nor tetragonal cells have magic harmonics even with circularly
polarized light. Through a careful group-theory analysis, we find that two
subgroups of symmetry operations unique to the hexagonal symmetry cancel out
third and ninth harmonics. This feature presents a rare opportunity to develop
HHG into a crystal-structure characterization tool for phase transitions
between hexagonal and nonhexagonal structures.
