Cluster-Based Typical-Medium Approaches to Anderson Localization in Two and Three Dimensions

Anderson localization (AL) in disordered media remains a central challenge for numerical simulation, especially at strong disorder where large system sizes and careful scaling are required. We study AL using two recent cluster-embedding frameworks: a) the real-space cluster extension of typical medium theory [1] and b) the momentum-space typical-medium dynamical cluster approximation (TMDCA) [2]. Applied to the three-dimensional Anderson tight-binding model with box and binary disorder, both methods capture localization across weak-to-strong disorder as the cluster size Nc increases, including the vanishing of the typical density of states at the mobility edge. The two approaches exhibit different Nc convergence behavior: the real-space cluster method converges at smaller Nc, whereas TMDCA slightly overestimates the critical disorder Wc. We further extend both methods to two dimensions, where strong nonlocal spatial correlations are essential for a correct description of AL, and compare accuracy and efficiency as functions of Nc. These results establish cluster-based typical-medium embeddings as practical tools for Anderson localization and lay the groundwork for materials-specific applications.

[1] K. M. Tam et al., “Real Space Real Space Quantum Cluster Formulation for the Typical Medium Theory of Anderson Localization”, Crystals 11 (11), 1282 (2021).

[2] H. Terletska et al., “Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems”, Appl. Sci., 8(12), 2401 (2018).