Anomalous mirror symmetry breaking in a model insulating cuprate Sr$_2$CuO$_2$Cl$_2$

Understanding the complex phase diagram of cuprate superconductors is an outstanding challenge. The most actively studied questions surround the nature of the pseudogap and strange metal states and their relationship to superconductivity. In contrast, there is general agreement that the low energy physics of the Mott insulating parent state is well captured by a two-dimensional spin $S$ = 1/2 antiferromagnetic (AFM) Heisenberg model. However, recent observations of a large thermal Hall conductivity in several parent cuprates appear to defy this simple model and suggest proximity to a magneto-chiral state that breaks all mirror planes perpendicular to the CuO$_2$ layers. Here we use optical second harmonic generation to directly resolve the point group symmetries of the model parent cuprate Sr$_2$CuO$_2$Cl$_2$. We report evidence of an order parameter $Phi$ that breaks all perpendicular mirror planes and is consistent with a magneto-chiral state in zero magnetic field. Although $Phi$ is clearly coupled to the AFM order parameter, we are unable to realize its time-reversed partner ($-Phi$) by thermal cycling through the AFM transition temperature ($T_{textrm{N}}$ $approx$ 260 K) or by sampling different spatial locations. This suggests that $Phi$ onsets above $T_{textrm{N}}$ and may be relevant to the mechanism of pseudogap formation.