A systematic approximation method for disordered systems

Abstract

Within a suitable generalisation of the renormalisation transformation group developed by Bogoliubov-Shirkov, the subsequent Lie differential equations prove to describe averaged behaviours of disordered systems precisely. In this sense, they may be looked upon as the fundamental equations for the system considered. By expanding the differential equations in power series of the invariant coupling and truncating them at an appropriate order, the authors can establish a systematic approximation method. The scheme is automatically assured to be valid in the weak-coupling case and turns out to have a well-defined limit of applicability. Moreover, it can be shown that the scheme is compatible with the Pauli exclusion principle and that the solutions obtained are, within the range of applicability, free from off-real-axis singularities in the complex energy plane. The lowest and the next order solutions are shown to correspond respectively to the virtual crystal and the average t-matrix approximation. The authors treat in this paper the statistically averaged behaviour of electrons in alloys with site-diagonal and non-correlated randomness. The extension to the case where there exist interactions between substitutional impurities is straightforward.