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The effect of disorder on lattice vibrational modes has been a topic of interest for several decades. In this work, we employ a Green's function based approach, namely, the dynamical cluster approximation (DCA), to investigate phonons in mass-disordered systems. Detailed benchmarks with previous exact calculations are used to validate the method in a wide parameter space. An extension of the method, namely, the typical medium DCA (TMDCA), is used to study Anderson localization of phonons in three dimensions. We show that, for binary isotopic disorder, lighter impurities induce localized modes beyond the bandwidth of the host system, while heavier impurities lead to a partial localization of the low-frequency acoustic modes. For a uniform (box) distribution of masses, the physical spectrum is shown to develop long tails comprising mostly localized modes. The mobility edge separating extended and localized modes, obtained through the TMDCA, agrees well with results from the transfer matrix method. A reentrance behavior of the mobility edge with increasing disorder is found that is similar to, but somewhat more pronounced than, the behavior in disordered electronic systems. Our work establishes a computational approach, which recovers the thermodynamic limit, is versatile and computationally inexpensive, to investigate lattice vibrations in disordered lattice systems.

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Physical Review B