Experimental quantification of electronic symmetry breaking through orbital hybridization phase

Symmetry classification of crystal structures has been central to predicting physical properties of materials. While such structural classification identifies which physical responses are symmetry-allowed, the magnitudes of these responses are governed by the degree of symmetry breaking in the electronic state. However, a well-defined quantitative descriptor for the electronic symmetry breaking has been established only in limited cases such as electric polarization and magnetization. No analogous descriptor exists for most other types, including chirality. Here, we propose an experimental framework for quantifying electronic symmetry breaking from the anisotropy of valence electron density distribution. We show that the orbital hybridization phases governing this anisotropy can be uniquely determined under site symmetry constraints. Applying this framework to structurally chiral transition-metal silicides, we determine hybridization phases from their valence electron densities observed by synchrotron X-ray diffraction. From the obtained complex hybridization, we quantify an electronic chirality $χ$ and theoretically demonstrate that it is directly proportional to circular dichroism, establishing $χ$ as a predictive descriptor of chiral responses. This approach is systematically applicable to various point groups, offering a general route to quantifying electronic symmetry breaking and predicting associated physical properties.