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arXiv:0906.2052 · Physical Review Letters

Many-Body Green Function of Degenerate Systems

Christian Brouder, Gianluca Panati, Gabriel Stoltz

Published Dec 1, 2009• 20 citations
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Abstract

A rigorous nonperturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H(0) leading to eigenstates of H(0) + V for degenerate systems. These initial states are eigenstates of P(0)VP(0), where P(0) is the projection onto a degenerate eigenspace of H(0). This result is used to give the proper definition of the Green function, the statistical Green function and the nonequilibrium Green function of degenerate systems. The convergence of these Green functions is established.

DOI
10.1103/physrevlett.103.230401

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A rigorous nonperturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H(0) leading to eigenstates of H(0) + V for degenerate systems. These initial states are eigenstates of P(0)VP(0), where P(0) is the projection onto a degenerate eigenspace of H(0). This result is used to give the proper definition of the Green function, the statistical Green function and the nonequilibrium Green function of degenerate systems. The convergence of these Green functions is established.
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.
We study a simple system described by a $2\ifmmode\times\else\texttimes\fi{}2$ Hamiltonian and the evolution of its quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate, we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows the evolution of eigenstates to be followed. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.
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.
This review deals with the state of the art and perspectives of description of nonequilibrium many-body systems using the nonequilibrium Green's function (NGF) method. The basic aim is to describe time evolution of the many-body system from its initial state over its transient dynamics to its long time asymptotic evolution. First, we discuss basic aims of transport theories to motivate the introduction of the NGF techniques. Second, this article summarizes the present view on construction of the electron transport equations formulated within the NGF approach to nonequilibrium. We discuss incorporation of complex initial conditions to the NGF formalism, and the NGF reconstruction theorem, which serves as a tool to derive simplified kinetic equations. Three stages of evolution of the nonequilibrium, the first described by the full NGF description, the second by a non-Markovian generalized master equation and the third by a Markovian master equation will be related to each other.
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
This review deals with the state of the art and perspectives of description of nonequi-librium many-body systems using the nonequilibrium Green’s function (NGF) method. The basic aim is to describe time evolution of the many-body system from its initial state over its transient dynamics to its long time asymptotic evolution. First, we discuss basic aims of transport theories to motivate the introduction of the NGF techniques. Second, this article summarizes the present view on construction of the electron transport equations formulated within the NGF approach to nonequilibrium. We discuss incorporation of complex initial conditions to the NGF formalism, and the NGF reconstruction theorem, which serves as a tool to derive simplified kinetic equations. Three stages of evolution of the nonequilibrium, the first described by the full NGF description,the second by a non-Markovian generalized master equation and the third by a Markovianmaster equation will be related to each other.