Nonlinear and time-resolved optical study of the 112-type iron-based superconductor parent Ca$_{1-x}$La$_{x}$FeAs$_{2}$ across its structural phase transition

The newly discovered 112-type ferropnictide superconductors contain chains of As atoms that break the tetragonal symmetry between the $a$ and $b$ axes. This feature eliminates the need for uniaxial strain that is usually required to stabilize large single domains in the electronic nematic state that exists in the vicinity of magnetic order in the iron-based superconductors. We report detailed structural symmetry measurements of 112-type Ca$_{0.73}$La$_{0.27}$FeAs$_{2}$ using rotational anisotropy optical second harmonic generation. This technique is complementary to diffraction experiments and enables a precise determination of the point group symmetry of a crystal. By combining our measurements with density functional theory calculations, we uncover a strong optical second harmonic response of bulk electric dipole origin from the Fe and Ca $3d$-derived states that enables us to assign $C_2$ as the crystallographic point group. This makes the 112-type materials high-temperature superconductors without a center of inversion, allowing for the possible mixing of singlet and triplet Cooper pairs in the superconducting state. We also perform pump-probe transient reflectivity experiments that reveal a 4.6 THz phonon mode associated with the out-of-plane motion of As atoms in the FeAs layers. We do not observe any suppression of the optical second harmonic response or shift in the phonon frequency upon cooling through the reported monoclinic-to-triclinic transition at 58 K. This allows us to identify $C_1$ as the low-temperature crystallographic point group but suggests that structural changes induced by long-range magnetic order are subtle and do not significantly affect electronic states near the Fermi level.