The mean-field approximation to the many-body Green's function (E-H${)}^{\mathrm{\ensuremath{-}}1}$ is derived by the use of products of single particle orbitals for trial functions in a variational formalism. It is shown that the matrix elements of the Green's function behave smoothly as ImE is varied towards the on-shell limit ImE=0.
The mean-field approximation to the many-body Green's function (E-H)\ensuremath−1 is derived by the use of products of single particle orbitals for trial functions in a variational formalism. It is shown that the matrix elements of the Green's function behave smoothly as ImE is varied towards the on-shell limit ImE=0.
The exact study of many-body microscopic systems is impossible when the number of particles is large (N ≥ 3). Approximate methods are then used. The time-independent mean-field (TIMF) approximation has been proposed for the description of collisions in many-body systems. Collision amplitudes are derived by the use of a variational principle and the choice of trial functions as products of single-particle orbitals. Resulting mean-field equations with a nonvanishing right-hand side turn out to be a generalization of the traditional Hartree or Hatree–Fock type equations. These TIMF equations are successfully solved numerically for the case of short-range forces. In this paper, we test the validity of this theory for the Coulomb interaction between two particles, that is, a long-range interaction. A numerical comparison between the exact and the mean-field solutions is conducted PACS Nos.: 31.15.Ne, 31.15.Pf, 21.45.+v,25.10.-i
The time-independent mean-field theory of collisions converts an intractable linear problem, namely, the calculation of the many-body propagator (E-H${)}^{\mathrm{\ensuremath{-}}1}$, into a numerically solvable problem, namely, the coupled calculation of single-particle propagators (\ensuremath{\eta}-h${)}^{\mathrm{\ensuremath{-}}1}$. The latter problem, however, reflects its nonlinearity through many solutions, out of which only one is nonspurious and gives a good approximation to the unique solution (off shell) of the former, linear problem. We prove the stability of this nonspurious solution in the special case of a four-body, one-dimensional collision. The experience thus gained is used to generate transition-shell models in the case of a realistic, three-dimensional, four-body elastic collision.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
The time‐dependent Green’s functions formalism provides a consistent description of the time evolution of quantum many‐body systems, either in the mean‐field approximation or in more sophisticated correlated approaches. We describe an attempt to apply this formalism to the mean‐field dynamics of symmetric reactions for one‐dimensional nuclear slabs. We pay particular attention to the off‐diagonal elements of the Green’s functions in real space representation. Their importance is quantified by means of an elimination scheme based on a super‐operator cut‐off field and their relevance for the global time evolution is assessed. The Wigner function and its structure in the mean‐field approximation is also discussed.
The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic resummation techniques and to nonperturbative dynamical mean field theory in particular. Contrary to functionals of the Green's function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.
Exact solutions to the Schrödinger equation for realistic nanoscale systems are beyond reach, hence, different strategies for approximating the solutions are necessary. Perturbation theory relies on a “zero-order Hamiltonian” to express the desired eigenvalues and eigenvectors of “the full Hamiltonian.” We derive working equations for the Rayleigh–Schrödinger perturbation theory, and the validity of the approach is analyzed for a generic two-level system. Applications are given to atoms perturbed by point charges or static fields, and for electrons in quantum wells. An alternative strategy is the variation method, which replaces exact solutions by their projection on a reduced space of “trial functions,” varied to minimize the associated error. Particularly important is the method of linear variation, which can potentially converge to the exact Hamiltonian eigenstates. The mean-field approximation, commonly used for many-particle systems, is derived by optimizing a trial function in the form of product of single-particle functions.