  One of the long-standing issues concerning the thermal properties of
amorphous solids is the complex pattern of phonon transport. Recent advances in
experiments and computer simulations have indicated a crossover from Rayleigh
scattering to viscous damping. A number of theories have been proposed, yet
critical tests are missing and the validity of these theories is unclear. In
particular, the precise location of the crossover frequency remains
controversial, and more critically, even the validity of the Rayleigh
scattering itself has been seriously questioned. The present work settles these
issues by using information on the underlying vibrational eigenmodes.
Vibrational eigenmodes were recently clarified over a wide frequency regime: a
mixture of phonon modes and soft localized modes in the continuum limit and
disordered and extended modes in the boson peak regime. Here, we demonstrate
that Rayleigh scattering occurs in the continuum limit and viscous damping
occurs in the boson peak regime, and these behaviors are therefore linked to
the underlying eigenmodes in the corresponding frequency regimes. Our results
unambiguously determine the crossover frequency. Furthermore, we establish
characteristic scaling laws of phonon transport, which are consistent with the
prediction of the mean-field theory at higher frequencies but inconsistent in
the low-frequency, Rayleigh scattering regime. Our results therefore reveal
crucial issues to be solved with regard to the current theory.
