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arXiv:cond-mat/0312158

Many-body theory of degenerate systems: A simple example

Christian Brouder

Published Dec 5, 2003• 1 citation
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Abstract

The hierarchy of Green functions for (quasi)degenerate systems, presented in cond-mat/0308058, is calculated in detail for the case of a system with closed shells plus a single electron in a two-fold degenerate level. The complete hierarchy is derived explicitly, in terms of Green functions and of Feynman diagrams.
DOI
10.48550/arxiv.cond-mat/0312158

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To set up a self-consistent quantum field theory of degenerate systems, the unperturbed state should be described by a density matrix instead of a pure state. This increases the combinatorial complexity of the many-body equations. Hopf algebraic techniques are used to deal with this complexity and show that the Schwinger-Dyson equations are modified in a non-trivial way. The hierarchy of Green functions is derived for degenerate systems, and the case of a single electron in a two-fold degenerate orbital is calculated in detail.
In many systems containing open shell ions, the wave function of the ions cannot be described by a single Slater determinant. This invalidates standard many-body theory and requires the use of nonequilibrium quantum field theory with initial correlations. Quantum group techniques are used here to deal with this combinatorially complex theory. The hierarchy of interacting Green functions for general initial correlations is derived and the case of a system with two-particle correlations is treated in detail.
\n \nIn many systems containing open shell ions, the wave function of\nthe ions cannot be described by a single Slater determinant. This\ninvalidates standard many-body theory and requires the use of\nnonequilibrium quantum field theory with initial correlations.\nQuantum group techniques are used here to deal with this\ncombinatorially complex theory. The hierarchy of interacting\nGreen functions for general initial correlations is derived and\nthe case of a system with two-particle correlations is treated in\ndetail.\n\n
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<).
A new approach to the two-time Green function method is proposed. In order to give a systematic way of decoupling the two-time Green functions and to overcome the difficulties often encountered in terminating the hierarchy of Green functions, a generating function is introduced and its properties are examined in detail for a simple model. It is shown that the generating function provides poweful tool even in the case where conventional approximation for Green functions does not work. An example of the generating function is constructed for Fermion system with two-body interaction.
In quantum field theory the Green function is usually calculated as the expectation value of the time-ordered product of fields over the vacuum. In some cases, especially in degenerate systems, expectation values over general states are required. The corresponding Green functions are essentially more complex than in the vacuum, because they cannot be written in terms of standard Feynman diagrams. Here a method is proposed to determine the structure of these Green functions and to derive nonperturbative equations for them. The main idea is to transform the cumulants describing correlations into interaction terms.