Coherent potential approximation for disordered bosons

A family of random models for bosonic quasi-particle excitations, e.g. the vibrations of a disordered solid, is introduced. The generator of the linearized phase space dynamics of these models is the sum of a deterministic and a random part. The former may describe any model of N identical phonon bands, while the latter is a d-dimensional generalization of the random matrix model of Lueck, Sommers, and Zirnbauer (LSZ). The models are constructed so as to exclude the unphysical occurrence of runaway solutions. By using the Efetov-Wegner supersymmetry method in combination with the new technique of superbosonization, the disordered boson model is cast in the form of a supermatrix field theory. A self-consistent approximation of mean-field type arises from treating the field theory as a variational problem. The resulting scheme, referred to as a coherent potential approximation, becomes exact for large values of N. In the random-matrix limit, agreement with the results of LSZ is found. The self-consistency equation for the full d-dimensional problem is solved numerically.