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Physical Review A

Closed-form intermediate representations of many-body propagators and resolvent matrices

J. Schirmer

Published May 1, 1991• 227 citations
Publisher

Abstract

A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.

DOI
10.1103/physreva.43.4647

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A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}\ifmmode~\else˜\fiJ{\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}\ifmmode~\else˜\fiJ​〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}\ensuremathΨn{\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}\ensuremathΨn​〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈\ensuremathΨn{\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}\ensuremathΨn​\ensuremath{\Vert}C{\mathrm{^}}_{\mathit{J}}\ensuremath{\Vert}\ensuremathΨ0N{\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}\ensuremathΨ0N​〉, where C{\mathrm{^}}_{\mathit{J}} denotes a physical excitation operator and \ensuremath{\Vert}\ensuremathΨ0N{\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}\ensuremathΨ0N​〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
A purely algebraic approach to Green's functions is presented. The new formulation allows the determination of a closed-form equation for Green's functions which depends only on the ground state of the system. By a straightforward perturbation expansion of this state an interesting approximation scheme is obtained which is nonperturbative in the final states (e.g., excited or ionic states). This method, which can be used for various propagators, is explicitly applied to the particle-particle propagator carrying out the scheme up to third order. The approach is discussed in detail and briefly compared with the well-established algebraic diagrammatic construction approximation scheme.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
In this paper, we develop a time-dependent approach to many-body perturbation theory for closed shells based on the resolvent of the Schrödinger equation. We introduce a quantity S(t) = i<φ‖exp [−i(H−E0)t]‖ψ≳/<φ‖ψ≳, where φ and ψ are, respectively, the unperturbed and exact wave functions for the system and E0 is the unperturbed energy. The Fourier transform of S(t), S(ω) = <φ‖(ω+E0− H)−1‖ψ≳/ <φ‖ψ≳, is a matrix element of the resolvent containing the exact function ψ and, thus, has a pole at ω = (E−E0), the correlation energy. Starting from a time-dependent perturbation expansion of S(t) via the Gellman–Low adiabatic theorem, we have obtained a Dyson-like equation: S−1(ω) = S−10(ω)+Σ<φ‖ψ≳ for S(ω). Such a derivation requires judicious grouping of terms of the perturbation series for S(t). It has been shown that specific regroupings of the terms of S(t) into appropriate ’’top’’ and ’’bottom’’ parts and corresponding time-integration procedures yield a Σ which generates the Brillouin–Wigner (BW) or Rayleigh–Schrödinger (RS) energy series. Σ for both the cases are identified as the ’’top’’ parts. The characteristic features of the BW series, namely, the appearance of disconnected and ω-dependent diagrams is to be contrasted with that of the RS series, which contains connected and ω-independent diagrams only, and both emerge naturally as a consequence of the dissection procedure into appropriate top and bottom parts.
The ground state and low excited states of liquid He3{\mathrm{He}}^{3}He3 (and other fermion systems) can be constructed from a set of basis functions $\ensuremath{\Psi}(|\mathrm{n})={{\ensuremath{\psi}}_{0}}^{B}\ensuremath{\Phi}(|\mathrm{n})$ in which ${{\ensuremath{\psi}}_{0}}^{B}$ is the ground-state boson-type solution of the Schr\"odinger equation and the model functions $\ensuremath{\Phi}(|\mathrm{n})$ are Slater determinants suitable for describing states of the noninteracting Fermion system. Diagonal and nondiagonal matrix elements of the identity and the Hamiltonian operator are evaluated by a cluster-expansion technique. An orthonormal basis system is constructed from $\ensuremath{\Psi}(|\mathrm{n})$ and used to express the Hamiltonian operator in quasiparticle form: a large diagonal component containing constant, linear, quadratic, and cubic terms in free-quasiparticle occupation-number operators and a nondiagonal component representing the residual interactions involved in collisions of two and three free quasiparticles.
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 .
A formulation of the propagator for a quantum system is constructed to simplify computation for multiparticle systems. The propagator is manifested in terms of any complete set on the Hilbert space of physical states of the system. In the short-time limit a particularly simple matrix expression is obtained using an energy eigenstate basis. This formulation is shown to contain all orders of perturbation theory in an approximate form. Identical particle exchange is handled in a straightforward manner in which only the number of terms, but not the complexity, of expressions is affected. The number of terms is held to manageable proportions by approximating multiple-particle exchange with repeated operations containing only two-particle exchange. Three-particle exchange is represented exactly by this approximation in the limit of infinitesimal time steps. An argument is presented that indicates that this is also true for an arbitrary particle number.