A typical medium cluster approach for multi-branch phonon localization

TL;DR

This work addresses how the vector, three-branch nature of phonons influences Anderson localization in disordered lattices by extending the typical medium dynamical cluster approximation (TMDCA) to multi-branch phonons. The authors develop a Green's-function based multi-branch DCA/TMDCA framework with mass disorder, validate it against exact diagonalization and limiting cases, and show that the typical density of states (TDOS) serves as an effective order parameter for localization while the arithmetic DOS (ADOS) fails to distinguish localized from extended states. They find that inter-branch couplings have minimal qualitative impact on the Anderson transition for the studied model, and they demonstrate accurate mobility-edge trajectories under box disorder with finite cluster sizes (e.g., $N_c=4^3$). The methodology provides a computationally efficient route to explore phonon localization in real materials and complex geometries, enabling first-principles-informed control of thermal transport via localization phenomena, with potential applications to layered structures and interfaces. In short, the multi-branch DCA/TMDCA framework advances phonon localization theory by incorporating vector phonons and confirming robust localization behavior across coupling regimes, while remaining tractable for realistic systems.

Abstract

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. In order 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 multi-branch 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 multi-branch case shows that the vector nature of the phonons does not affect the Anderson transition of phonons significantly. The developed multi-branch TMDCA formalism can be employed for studying phonon localization in real materials.

Figures (9)

• Figure 1: Parameters: $\tau=0.3, c_A=0.2,V_A=0.9$. A comparison of the ADOS obtained by the developed multi-branch DCA, single site CPA and the exact diagonalization method (ED). Panel a) shows the ADOS obtained from the multi-branch DCA using a cluster size of $N_c=4^3$ and the ED results for $\varepsilon=0$. Panel b) compares the single-site CPA results with ED results for $\varepsilon=0$. Note that, unlike the DCA, the CPA fails to reproduce the fine-structure features of the high-frequency impurity mode of the ADOS as found in exact calculations. Panels c) and d) show the multi-branch DCA results for the case where the atomic vibrations at lattice site $R_l$ in the direction $\alpha$ are coupled with atomic vibrations around $R_{l'}$ in the direction $\beta$ by the tuning parameter $\varepsilon$ of $0.2$ and $0.4$, respectively. It is observed that the host mode height changes with increasing $\varepsilon$, and the multi-branch DCA results are in strong agreement with the ED results.
• Figure 2: Parameters: $\tau=0.3, \varepsilon=0.0, c_A=0.2, V_A=0.9$. Left panel: comparison of the ADOS obtained from Eq. \ref{['eq:dos_branch_relation']} and from the multi-branch DCA formalism. Right panel: comparison of the TDOS obtained from Eq. \ref{['eq:tdos_branch_relation']} and the multi-branch TMDCA formalism.
• Figure 3: Parameters: $\tau=0.3, c_A=0.2, V_A=0.9, N_c=4^3.$ Panels a), b), and c) show the evolution of the average ADOS and the typical TDOS for increasing values of $\varepsilon=0,0.2,0.4$ at at large disorder ($V_A=0.9$) and small impurity concentration $c_A=0.2$. Panel d) shows the ratio of TDOS/ADOS for all values of $\varepsilon=0,0.2,0.4$.
• Figure 4: Parameters: $\tau=0.3, c_A=0.5, V_A=0.9, N_c=4^3$. Panels a), b), and c) show the evolution of the average ADOS and the typical TDOS for increasing values of $\varepsilon=0,0.2,0.4$ at large disorder ($V_A=0.9$) and moderate impurity concentration ($c=0.5$). d) the TDOS/ADOS ratio at $\varepsilon=0.0, 0.2, 0.4$.
• Figure 5: Parameters: $\tau=0.3, c_A=0.2, V_A=0.5, N_c=4^3$. Panels (a), (b), and (c) show the evaluation of ADOS) and TDOS and with increasing $\varepsilon$ values from 0 to 0.2 to 0.4 at intermediate disorder strength ($V_A=0.5$). (d) the TDOS/ADOS ratio at $\varepsilon=0.0, 0.2, 0.4$