Implications of second harmonic generation for hidden order in Sr${}_2$CuO${}_2$Cl${}_2$

Sr${}_2$CuO${}_2$Cl${}_2$ (SCOC) is a model undoped cuprate with $I4\mathrm{/}mmm$ crystallographic symmetry, and a simple magnetic space group $C_{A}mca$ with associated magnetic point group $mmm{1}^{\mathrm{\prime }}$. However, recent second harmonic spectroscopy in the antiferromagnetic phase has challenged this picture, suggesting instead a magnetic point group $4\mathrm{/}m{m}^{\mathrm{\prime }}{m}^{\mathrm{\prime }}$ that co-exists with the antiferromagnetism and breaks the two orthogonal mirror planes containing the tetragonal $c$-axis. Here, we analyze the symmetry of SCOC in light of the second harmonic results, and discuss possible ground states that are consistent with the data.