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Journal of Physics C Solid State Physics

A band model for the understanding of divergences in perturbation theory

P Erdos, András Sütö

Published Jul 10, 1984• 0 citations
Publisher

Abstract

A simple theoretical model is constructed on the basis of real many-body systems occurring in solids. One example is an impurity in a host with band structure. The model allows an exact solution to be obtained as well as approximations by perturbational methods such as the Rayleigh-Schrodinger and Brillouin-Wigner techniques. The cause and nature of the divergence of perturbation theory is demonstrated and the utility of partial sums of divergent perturbation series is explained.

DOI
10.1088/0022-3719/17/19/009

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A simple theoretical model is constructed on the basis of real many-body systems occurring in solids. One example is an impurity in a host with band structure. The model allows an exact solution to be obtained as well as approximations by perturbational methods such as the Rayleigh-Schrodinger and Brillouin-Wigner techniques. The cause and nature of the divergence of perturbation theory is demonstrated and the utility of partial sums of divergent perturbation series is explained.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
Perturbation methods are generally used for solving wave operator equations associated with the determination of effective Hamiltonians.In many cases the standard Rayleigh-Schrödinger and Brillouin-Wigner series either converge slowly or diverge.Therefore it is necessary to modify or to renormalize the standard wave equations.For that purpose derivative and convergence superoperators within the Ralyeigh-Schrödinger and Brillouin-Wigner formalisms were introduced.A new efficient approximation for convergence superoperators is investigated in this paper.Its application to a model system of N non-interacting molecules shows that this approximation can overcome convergence difficulties.