  It is well known that amorphous solids display a phonon spectrum where the
Debye $\sim \omega^2$ law at low frequency melds into an anomalous excess-mode
peak (the boson peak) before entering a quasi-localized regime at higher
frequencies dominated by scattering. The microscopic origin of the boson peak
has remained elusive despite various attempts to put it in a clear connection
with structural disorder at the atomic/molecular level. Using numerical
calculations on model systems, we show that the microscopic origin of the boson
peak is directly controlled by the local breaking of center-inversion symmetry.
In particular, we find that both the boson peak and the nonaffine softening of
the material display a strong positive correlation with a new order parameter
describing the local inversion symmetry of the lattice. The standard
bond-orientational order parameter, instead, is shown to be a poor correlator
and cannot explain the boson peak in randomly-cut crystals with perfect
bond-orientational order. Our results bring a unifying understanding of the
boson peak anomaly for model glasses and defective crystals in terms of a
universal local symmetry-breaking principle of the lattice.
