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arXiv:hep-th/0307212

Quantum field theory of degenerate systems

Christian Brouder

Published Jul 22, 2003• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.hep-th/0307212

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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.
The many-body theory of degenerate systems is described in detail. More generally, this theory applies to systems with an initial state that cannot be described by a single Slater determinant. The double-source (or closed-time-path) formalism of nonequilibrium quantum field theory is used to derive an expression for the average value of a product of interacting fields when the initial state is not the vacuum or a Slater determinant. Quantum group techniques are applied to derive the hierarchy of unconnected Green functions and the hierarchy of connected ones.
An ensemble Green's function formalism, based on the von Neumann density matrix approach, to calculate one-electron excitation spectra of a many-electron system with degenerate ground states is proposed. A set of iterative equations for the ensemble Green's function and self-energy is derived and a simplest approximation corresponding to an ensemble GW\mathit{GW}GW approximation is naturally obtained. The derivation is based on the Schwinger functional derivative technique and does not assume any adiabatic connection between a noninteracting and an interacting ground state.
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.
The density matrix of a nonrelativistic quantum system, divided into NNN subsystems, is rewritten in terms of the set of all partitioned density matrices for the system. For the case where the different subsystems are distinguishable, we derive a hierarchy of equations of motion linking the dynamics of all the partitioned density matrices, analogous to the ``Schwinger-Dyson'' hierarchy in quantum field theory. The special case of a set of NNN coupled spin-1/21/21/2 ``qubits'' is worked out in detail. The equations are then rewritten in terms of a set of ``entanglement correlators,'' which comprises all the possible correlation functions for the system---this case is worked out for coupled spin systems. The equations of motion for these correlators can be written in terms of a first-order differential equation for an entanglement correlator supervector.
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.