  Amorphous solids have excess soft modes in addition to the phonon modes
described by the Debye theory. Recent numerical results show that if the phonon
modes are carefully removed, the density of state of the excess soft modes
exhibit universal quartic scaling, independent of the interaction potential,
preparation protocol, and spatial dimensions. We hereby provide a theoretical
framework to describe this universal scaling behavior. For this purpose, we
extend the mean-field theory to include the effects of finite dimensional
fluctuation. Based on a semi-phenomenological argument, we show that mean-field
quadratic scaling is replaced by the quartic scaling in finite dimensions.
Furthermore, we apply our formalism to explain the pressure and protocol
dependence of the excess soft modes.
