A new method to treat the energy dissipation into electron excitations by time-dependent local perturbations is presented. The probability P( epsilon ) for the electron system to be excited to a state epsilon is expressed in terms of new Green functions given by extending the Keldysh Green function formalism. As an example, the method is applied to the slowly varying noninteracting Wolff model at zero temperature. The Muskhelishvili technique, which has been used for X-ray problems, is used to solve the resulting integral equations of the Green functions. The results are identical to those given by a time-dependent Tomonagon approach and by solving directly the equation of motion for field operators.
A new method to treat the energy dissipation into electron excitations by time-dependent local perturbations is presented. The probability P( epsilon ) for the electron system to be excited to a state epsilon is expressed in terms of new Green functions given by extending the Keldysh Green function formalism. As an example, the method is applied to the slowly varying noninteracting Wolff model at zero temperature. The Muskhelishvili technique, which has been used for X-ray problems, is used to solve the resulting integral equations of the Green functions. The results are identical to those given by a time-dependent Tomonagon approach and by solving directly the equation of motion for field operators.
The Green function formalism introduced by Keldysh for irreversible processes is applied to the calculation of the time-dependent charge exchange probabilities in scattering problems. Considering the initial state of two non-interacting subsystems described by an independent particle model, their interaction is taken in a first stage, as evolving in time according to a time-dependent Hartree - Fock (TDHF) scheme. The equations of motion for the two-times Green functions are subsequently obtained in Dyson-like form by including the residual electronic correlations expanded perturbatively up to second order. It is shown that, by solving these equations of motion, this in turn allows one to calculate the two-particle correlation functions at equal times, as required to describe the fractions of charge in each subsystem. This procedure contrasts with that of considering expectation values of the correlation functions taken between the corrected TDHF state up to an equivalent order in the perturbation. In order to test the advantages of this method we have applied this scheme to the scattering of an atom from a three-substrate atom chain described by an Anderson Hamiltonian, where comparisons with exact solutions can be easily established. We found that, in this case, the fractions of charges carried by the scattered particle obtained with our proposal compare fairly well with exact results, within an ample range of parameter selection.
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.
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).
The zero-temperature single-particle Green's function of correlated fermion models with moderately large Hilbert-space dimensions can be calculated by means of Krylov-space techniques. The conventional Lanczos approach consists of finding the ground state in a first step, followed by an approximation for the resolvent of the Hamiltonian in a second step. We analyze the character of this approximation and discuss a numerically exact variant of the Lanczos method which is formulated in the time domain. This method is extended to obtain the nonequilibrium single-particle Green's function defined on the Keldysh-Matsubara contour in the complex time plane which describes the system's nonperturbative response to a sudden parameter switch in the Hamiltonian. The proposed method will be important as an exact-diagonalization solver in the context of self-consistent or variational cluster-embedding schemes. For the recently developed nonequilibrium cluster-perturbation theory, we discuss its efficient implementation and demonstrate the feasibility of the Krylov-based solver. The dissipation of a strong local magnetic excitation into a non-interacting bath is considered as an example for applications.
We discuss microscopic foundations of dissipation arising in a model Fermi or Bose system with weak local interaction. We consider the dynamics of equilibrium fluctuations in the Keldysh–Schwinger formalism and discuss the relation between dissipation and pinch singularities of perturbation theory diagrams. Using the Dyson equation, we define and calculate the dissipation parameter in the two-loop approximation. We show that this parameter is analogous to Onsager’s kinetic coefficient and is associated with decay in the quasiparticle spectrum.
This chapter examines the energy of the resulting quasi-particle using the Green function technique. The field-theoretic Green function technique developed originally by Feynman, Dyson, and others for quantum electrodynamics finds extensive applications to the many body problems encountered in different branches of physics, for its proven efficacy in dealing with the dynamical effects associated with these problems. The Green function formalism is essentially a systematic time-dependent perturbation theory involving certain quantities called the Green functions or propagators. The energy-dependent Green function, G, provides great many physical information about a system, in particular the energy of the quasi-particles. Puff and Whitfield have considered the thermodynamic Green functions for the problem and found at the end of the calculation the zero-temperature limit. In the zero-temperature limit the approximate thermodynamic Green function of Mahan becomes equivalent to the intermediate coupling polaron model of Haeringen and is thus expected to give the same result as given by method of Haeringen.