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Canadian Journal of Physics

Moment Comparison of Two Decoupling Procedures for Green Function Equations

Barry Frank

Published Mar 15, 1971• 0 citations
Publisher

Abstract

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.

DOI
10.1139/p71-092

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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.
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.
The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and zzz nearest neighbors.
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.
The critical and tricritical behavior of the Takagi model is investigated using the method of double-time Green's functions. A Callen-type decoupling procedure appropriate to the paramagnetic phase is used. The critical curve in field space and the critical values of the density and average energy of the system are determined for the fcc lattice in the limit of equal numbers of bosons and vacancies. The tricritical point is located and the tricritical exponents are found to have classical values. These results are shown to be in good agreement with the series analysis of the model. The mean-field and random-phase-approximation (RPA) decoupled Green's-function theories are also discussed. The RPA treatment does not display a tricritical point.
A generalized random-phase approximation (RPA) for decoupling the double-time Green's functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.