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Theoretical and Mathematical Physics

Decoupling of two-time Green's functions and diagrammatic analysis

Nikolay M. Plakida

Published Oct 1, 1970• 26 citations
Publisher
DOI
10.1007/bf01035987

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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.

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The diagram technique, which was developed to solve a hierarchy of equations of motion for the reduced density matrices, is applied in order to solve a hierarchy of the equations of motion for the two-time Green's functions. The diagrammatical representations of the two-time Green's functions are obtained either in terms of the equilibrium one- and many-particle reduced density matrices or in terms only of the equilibrium one-particle reduced density matrix. The latter representation for the one-particle two-time Green's function and the formula by which the reduced density matrix is calculated from the Green's function give us a set of coupled equations which determines both the reduced density matrix and the two-time Green's function. The calculation with the aid of the present technique is illustrated for an example of Hubbard's calculation for the “Hubbard Hamiltonian”.

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The diagrammatic Monte Carlo (DiagMC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte Carlo method, and we term the improved version the "flat histogram diagrammatic Monte Carlo" method. We demonstrate the superiority of this method over the standard DiagMC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem.

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The Two-time green functions for some two-level system

Tozaburo Otsu, Tadashi Watanebe

Physics reports of the Kumamoto University•Dec 20, 1982•0 citations
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.
The diagram technique, which was developed to solve a hierarchy of equations of motion for the reduced density matrices, is applied in order to solve a hierarchy of the equations of motion for the two-time Green's functions. The diagrammatical representations of the two-time Green's functions are obtained either in terms of the equilibrium one- and many-particle reduced density matrices or in terms only of the equilibrium one-particle reduced density matrix. The latter representation for the one-particle two-time Green's function and the formula by which the reduced density matrix is calculated from the Green's function give us a set of coupled equations which determines both the reduced density matrix and the two-time Green's function. The calculation with the aid of the present technique is illustrated for an example of Hubbard's calculation for the “Hubbard Hamiltonian”.
The diagrammatic Monte Carlo (DiagMC) method is a numerical technique which samples the entire diagrammatic series of the Green's function in quantum many-body systems. In this work, we incorporate the flat histogram principle in the diagrammatic Monte Carlo method, and we term the improved version the "flat histogram diagrammatic Monte Carlo" method. We demonstrate the superiority of this method over the standard DiagMC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any a priori knowledge about this function, by applying the technique to the polaron problem.