Evidence for a Parity Broken Monoclinic Ground State in the S = 1/2 Kagom\'{e} Antiferromagnet Herbertsmithite

Nearest-neighbor interacting S = 1/2 spins on the ideal Kagom\'{e} lattice are predicted to form a variety of novel quantum entangled states, including quantum spin-liquid (SL) and valence bond solid (VBS) phases. In real materials, the presence of additional perturbative spin interactions may further expand the variety of entangled states, which recent theoretical analyses show are identifiable through the spontaneous loss of particular discrete point group symmetries. Here we comprehensively resolve the ground state point group symmetries of the prototypical Kagom\'{e} SL candidate ZnCu$_3$(OH)$_6$Cl$_2$ (Herbertsmithite) using a combination of optical ellipsometry and wavelength-dependent multi-harmonic optical polarimetry. We uncover a subtle parity breaking monoclinic structural distortion at a temperature above the nearest-neighbor exchange energy scale. Surprisingly, the parity-breaking order parameter is dramatically enhanced upon cooling and closely tracks the build-up of nearest-neighbor spin correlations, suggesting that it is energetically favored by the SL state. The refined low temperature symmetry group greatly restricts the number of viable ground states, and, in the perturbative limit, points toward the formation of a nematic $Z_2$ striped SL ground state - a SL analogue of a liquid crystal.