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Advances In Physics

Expressing the decoupled equations of motion for the Green's function as a partial sum of Feynman diagrams

R. D. Mattuck, Alba Theumann

Published Nov 1, 1971• 17 citations
Publisher

Abstract

It is shown that any given decoupling of the equations of motion for the propagators of an interacting Fermi system is equivalent to a particular partial sum of Feynman diagrams. This result is established for many-time propagators in a straightforward way by iterating in the diagrammatic form of the equations of motion. The iteration method does not work for two-time propagators, but we are able to present an alternative argument which makes it plausible that the result holds true in this case as well. The decouplings leading to the Hartree-Fock, ladder, and ring diagram (RPA) partial sums are discussed in detail. It is conjectured that the reverse correspondence is not true, i.e. to any given partial sum there does not necessarily correspond a decoupling.

DOI
10.1080/00018737100101331

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It is shown that any given decoupling of the equations of motion for the propagators of an interacting Fermi system is equivalent to a particular partial sum of Feynman diagrams. This result is established for many-time propagators in a straightforward way by iterating in the diagrammatic form of the equations of motion. The iteration method does not work for two-time propagators, but we are able to present an alternative argument which makes it plausible that the result holds true in this case as well. The decouplings leading to the Hartree-Fock, ladder, and ring diagram (RPA) partial sums are discussed in detail. It is conjectured that the reverse correspondence is not true, i.e. to any given partial sum there does not necessarily correspond a decoupling.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
For a system of indistinguishable fermions with pairwise interactions, we consider renormalized, perturbative expansions for the three- and four-particle propagators (Green's functions) in terms of exact one-particle propagators, along with a pair interaction that may be bare (as it appears in the Hamiltonian) or dressed (polarized). Care is taken to provide a rigorous foundation for the diagrammatic representations of these perturbation series; in particular, it is demonstrated how each topologically distinct diagram represents an embedding (in three-dimensional space) of numerous separate terms in the perturbation series. Within a renormalized ladder approximation for the three- and four-particle propagators, we derive diagram weights (which differ from the two-particle diagram weights) in order to limit the perturbation series to topologically distinct, permutationally independent diagrams. These results lay the foundation for perturbative approximations that decouple the propagator and density matrix equations of motion.
The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
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.