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Journal of the Physical Society of Japan

Resolution of the Hierarchy of Many-Body Green's Functions

Shigeji Fujita

Published Nov 1, 1969• 12 citations
Publisher

Abstract

The rigorous evolution equations for the one-body Green's functions ( g > , g < ) describing an imperfect gas in a finite volume are not closed (an analogue of B–B–G–K–Y hierarchy for the many-body distribution functions). The analysis of these equations are made in terms of proper connected diagrams. It is shown that in the bulk limit: N (particle number)→∞, \(\varOmega\)(volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.

DOI
10.1143/jpsj.27.1096

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The rigorous evolution equations for the one-body Green's functions ( g > , g < ) describing an imperfect gas in a finite volume are not closed (an analogue of B–B–G–K–Y hierarchy for the many-body distribution functions). The analysis of these equations are made in terms of proper connected diagrams. It is shown that in the bulk limit: N (particle number)→∞, (\varOmega)(volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.
The rigorous evolution equations for the one-body Green's functions (g>,g<{g}^{>},{g}^{<}g>,g<) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, NNN (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (g>,g<{g}^{>},{g}^{<}g>,g<).
The rigorous evolution equation for the one-body density matrix n ( t ) describing a quantum system of interacting particles is not closed according to the investigations of Bogoliubov, Born, Green, Kirkwood, Yvon and others (B-B-G-K-Y hierarchy). The analysis of this equation is made in terms of connected diagrams. It is found that in the limits of large volume and time this equation can be reduced to a generalized Boltzmann equation which is a closed equation with respect to n ( t ) and which can rigorously describe linear and non-linear transport coefficients.
Abstract We analyze the correspondence between two formalisms describing an interacting Bose gas, namely the standard Feynman diagrammatic expansion on the one hand, and the hierarchy equations for the imaginary-time Green functions on the other hand. We show that the Hartree–Fock approximation, as well as its first corrections at low density derived by Baym et al (2001 Eur. Phys. J. B 24 107–24), can be equivalently formulated in both formalisms. Within the hierarchy approach, the two-body correlations are expressed at lowest order in the interactions in terms of the full one-body Green function. This sheds light on the physical content of the corrections to the Hartree–Fock theory. Moreover, this representation of can be extended to higher orders, opening the way to systematic calculations of further corrections to the ideal critical temperature when the density increases.
Many functions of physical interest are defined by means of recursion relations where a function of a given rank can be expressed in terms of the functions of lower rank. It is also usually possible to define approximation schemes which, with natural modifications, can be applied to evaluate any function of the hierarchy. Typical examples are density functions, propagators, and Green's functions, and the approximation approaches to them are often based on perturbation theory. The attempt of this work is to clarify the connection between the approximation schemes for propagators and the hierarchy of linking equations. The concept of scheme consistency is introduced and used to extend available approximation schemes. As a specific example the theory is worked out for the particle-particle propagator, a quantity describing the simultaneous removal of two particles from a many-particle system. It is expected that the derived working equations are of importance in the description of double-ionization processes where each of the two removed electrons originates from a different part of the system.
In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.