Real-space cluster dynamical mean-field approach to the Falicov-Kimball model: An alloy-analogy approach

It is long known that the best single-site coherent potential approximation falls short of describing Anderson localization. Here, we study a binary alloy disorder [or equivalently, a spinless Falicov-Kimball (FK)] model and construct a dominantly analytic cluster extension that treats intracluster $(1/d,\phantom{\rule{0.16em}{0ex}}d=$ spatial dimension) correlations exactly. We find that, in general, the irreducible two-particle vertex exhibits clear nonanalyticities before the band splitting transition of the Hubbard type occurs, signaling onset of an unusual type of localization at strong coupling. Using time-dependent response to a sudden local quench as a diagnostic, we find that the long-time wave-function overlap changes from a power-law to an anomalous form at strong coupling, lending additional support to this idea. Our results also imply such ``strong'' localization in the equivalent FK model, the simplest interacting fermion system.