Typical medium cluster approach for multibranch phonon localization

The phenomenon of Anderson localization in various disordered media has sustained significant interest over many decades. Specifically, the Anderson localization of phonons has been viewed as a potential mechanism for creating fascinating thermal transport properties in materials. However, despite extensive work, the influence of the vector nature of phonons on the Anderson localization transition has not been well explored. To achieve such an understanding, we extend a recently developed phonon dynamical cluster approximation (DCA) and its typical medium variant (TMDCA) to investigate spectra and localization of multibranch phonons in the presence of pure mass disorder. We validate the new formalism against several limiting cases and exact diagonalization results. A comparison of results for the single-branch versus multibranch case shows that the vector nature of the phonons does not significantly affect the Anderson transition of phonons. The developed multibranch TMDCA formalism can be employed for studying phonon localization in real materials.