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Journal of Mathematical Physics

Application of infinite order perturbation theory in linear systems. II. The frequency spectrum of disordered chains

Shiyu Wu

Published Jul 1, 1974• 19 citations
Publisher

Abstract

A formulation is presented such that the ensemble average of the diagonal elements of the Green's function for a one-dimensional disordered system can be calculated to arbitrary accuracy. The method is illustrated with an application to an isotopically disordered chain. An approximate solution to the frequency spectrum of a disordered binary chain is also discussed.

DOI
10.1063/1.1666776

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Progress of Theoretical Physics Supplement•Jan 1, 2000•6 citations
A formulation is presented such that the ensemble average of the diagonal elements of the Green's function for a one-dimensional disordered system can be calculated to arbitrary accuracy. The method is illustrated with an application to an isotopically disordered chain. An approximate solution to the frequency spectrum of a disordered binary chain is also discussed.
For a one dimensional half-infinite chain with diagonal disorder we calculated the ultimate at t→∞t\to\inftyt→∞ value of the average excitation density at the edge site DDD if at t=0t=0t=0 the excitation was localised at the edge site (Anderson' s creterium). We obtained the following results: i) for the binary disordered chain we derived the close expression for DDD which is exact in the limit of low concentration of defects and is valid for an arbitrary energy of defects ε\varepsilonε. In this case DDD demonstrated the non analytical dependence on ε\varepsilonε. ii) The close expression for DDD is obtained for the case of an arbitrary small disorder. iii) The relative contribution of states with specified energy to DDD is calculated. All the results obtained are in complete agreement with computer simulation.
The disorder averaged single-particle Green's function of electrons subject to a time-dependent random potential with long-range spatial correlations is calculated by means of bosonization in arbitrary dimensions. For static disorder our method is equivalent with conventional perturbation theory based on the lowest order Born approximation. For dynamic disorder, however, we obtain a new non-perturbative expression for the average Green's function. Bosonization also provides a solid microscopic basis for the description of the quantum dynamics of an interacting many-body system via an effective stochastic model with Gaussian probability distribution.
An efficient scheme is introduced to calculate the Green function on the basis of polynomials expansion. This scheme has the following advantages: (i) it is possible to calculate both the diagonal and off-diagonal elements of the Green function, (ii) applicable to the system with discrete spectrum, (iii) integrated density of states can also be obtained by the same algorithm, and (iv) it is suitable for vector and parallel processing, and required CPU time and memory size are proportional only to the system size (i.e. ordered N). Applications are made to simple lattice systems for demonstration.