Analytic approximation for substitutional alloys

It is shown that the terms in the cumulant expansion of the coherent potential for an uncorrelated substitutional alloy can be combined by partial summation in such a way that the Herglotz property can be studied directly. A consistency condition for the Herglotz property is found, which is satisfied by the single-site coherent-potential approximation (CPA) but not by the $n$-site CPA for $n\ge 2$. A natural generalization of the CPA [referred to as the "traveling-cluster approximation" (TCA)], satisfying the consistency condition, is developed in which graphs involving arbitrarily many sites are involved, but in such a way that overlaps of cumulan averages involve only limited sets of sites. A fixed-point theorem is developed that guarantees that itera ion of the TCA equations for a broad range of physical systems converges to a unique self-consistent solution that preserves the Herglotz property of the mean resolvent. Calculations of the density of states using the nearest-neighbor TCA for a single-band tight-binding model are presented, and show a distinctly better fit to exact numerical results than the CPA, including some of the structure due to localized states.