Applying the linear δ expansion to disordered systems

We apply the linear d expansion ~LDE!, originally developed as a nonperturbative, analytical approximation scheme in quantum field theory, to problems involving noninteracting electrons in disordered solids. The initial idea that the LDE method might be applicable to disorder is suggested by the resemblance of the supersymmetric field theory formalism for quantities such as the disorder-averaged density of states and conductance to the path-integral expressions for the n-point functions of lf 4 field theory, where the LDE has proved a successful method of approximation. The field theories relevant for disorder have several unusual features that have not been considered before, however, such as anticommuting fields with Faddeev-Popov~FP! rather than Dirac-type kinetic-energy terms, imaginary couplings and Minkowskian field coordinate metric. Nevertheless we show that the LDE method can be successfully generalized to such field systems. As a preliminary test of the method and also to give some understanding of its origins, we calculate to third order in the LDE the ground-state energy of a supersymmetric anharmonic oscillator with FP kinetic term and real anharmonic coupling strength of arbitrary magnitude. Strong evidence for the convergence of the LDE is obtained. We then calculate to second order in the LDE the disorder-averaged density of states of a one-dimensional system and find even at first order more accurate results than the commonly used self-consistent Born approximation. In the final part we outline one possible way in which the LDE method might be applied to the conductance, using as supporting example a zero-dimensional model with Minkowskian field coordinate metric. Further directions for research are discussed in the conclusion. @S0163-1829~97!01439-2#