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The effect of disorder on lattice vibrational modes has been a topic ofinterest for several decades. In this work, we employ a Green's function basedapproach, namely the dynamical cluster approximation (DCA), to investigatephonons in mass disordered systems. Detailed benchmarks with previous exactcalculations are used to validate the method in a wide parameter space. Anextension of the method, namely the typical medium DCA (TMDCA), is used tostudy Anderson localization of phonons in three dimensions. We show that, forbinary isotopic disorder, lighter impurities induce localized modes beyond thebandwidth of the host system, while heavier impurities lead to a partiallocalization of the low-frequency acoustic modes. For a uniform (box)distribution of masses, the physical spectrum is shown to develop long tailscomprising mostly localized modes. The mobility edge separating extended andlocalized modes, obtained through the TMDCA, agrees well with results from thetransfer matrix method. A re-entrance behavior of the mobility edge withincreasing disorder is found that is similar to, but somewhat more pronouncedthan, the behavior in disordered electronic systems. Our work establishes a newcomputational approach, which recovers the thermodynamic limit, is versatileand computationally inexpensive, to investigate lattice vibrations indisordered lattice systems.

Localization of phonons in mass disordered alloys - A typical medium dynamical cluster approach