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Canadian Journal of Physics

A moment theorem for resonance line shapes arising from decoupled Green's function equations

Barry Frank, Robert Barrie

Published Apr 1, 1969• 2 citations
Publisher

Abstract

A theorem is derived which enables one to bypass quadrature in the calculation of moments of a resonance line obtained from any decoupling of the Green's function hierarchy of equations. These moments are those of the resonance line as expressed in terms of delta-function peaks prior to smearing. A comparison with exact theoretical moments then provides a test of the decoupling procedure as distinct from the smearing technique. The method is illustrated by application to the one-dimensional Ising model.

DOI
10.1139/p69-099

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A theorem is derived which enables one to bypass quadrature in the calculation of moments of a resonance line obtained from any decoupling of the Green's function hierarchy of equations. These moments are those of the resonance line as expressed in terms of delta-function peaks prior to smearing. A comparison with exact theoretical moments then provides a test of the decoupling procedure as distinct from the smearing technique. The method is illustrated by application to the one-dimensional Ising model.
A recently developed moment theorem and an approximation scheme of Pryce and Stevens are used to compare 2nd-RPA and Tomita–Tanaka type decoupling procedures for Green function equations. The comparison is accomplished from the point of view of the moments of the paramagnetic resonance line shapes that the decouplings generate. An Ising system in an external magnetic field is taken as the model system. It is found that use of the more complicated 2nd RPA is justified only if accuracy to O(1/z2) is desired.
Moment-conserving decoupling procedures for the one-electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment-conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self-consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two-moment scheme (Hartree-Fock) may be numerically closer to the exact answer than some three-moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle-hole propagator, appears to be the most consistent method.
A well-established method to deal with highly correlated systems is based on the expansion of the Green's function in terms of spectral moments. In the context of the Composite Operator Method one approximation is proposed: a set of n composite fields is assumed as fundamental basis and the dynamics is considered up to the order s. The resulting Green's function has a sn-pole structure. The truncation of the hierarchy of the equations of motion is made at the s-th order and the first s-1 equations are treated exactly. A theorem, which rules the conservation of the spectral moments, is presented. The procedure is applied to the Hubbard model and a recurrence relation for the calculation of its electronic spectral moments is derived.
The application of the Green function method to the Mollow problem is reconsidered. This problem consists of an intense monochromatic electromagnetic field in resonance with a two-level atom. We obtain a solution to the infinite hierarchy of Green function equations of motion instead of decoupling the equations of motion. This solution gives an excitation spectrum in agreement with Mollow's results. We conclude that earlier discrepancies in the excitation spectrum were artifacts of the decoupling procedure. Finally, we comment upon the physical ideas associated with the present and previous Green function treatments of the Mollow problem.