Spatially Dispersive Second-Harmonic Generation in Ferroaxial Systems

Spatially dispersive second-harmonic generation (SHG) provides a powerful probe of centrosymmetric multipolar states beyond the electric-dipole approximation. We develop a gauge-consistent microscopic theory of spatially dispersive SHG that treats electric-quadrupole (EQ) and magnetic-dipole (MD) processes on equal footing. Applying the formulation to a minimal triangular cluster model with ferroaxial order, which is closely related to an electric toroidal dipole, we show that the nonlinear optical response directly reflects the ferroaxial order parameter. The agreement between length- and velocity-gauge calculations confirms the gauge consistency of the formulation. We further demonstrate that the EQ and MD contributions exhibit different spectral weights despite sharing the same resonance energies. In particular, the MD channel can dominate over the EQ channel at selected resonances, indicating that it is not merely a perturbative correction. Our results establish that both EQ and MD processes are essential for a quantitative description of spatially dispersive SHG in ferroaxial materials and provide a microscopic basis for interpreting nonlinear optical signatures of centrosymmetric multipolar order.