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arXiv:1507.06407 · Physical Review B

Equation-of-motion series expansion of double-time Green's functions

Ning-Hua Tong

Published Oct 26, 2015• 8 citations
arXivHTMLPDF

Abstract

Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter $\ensuremath{\lambda}$ in the Hamiltonian $H={H}_{0}+\ensuremath{\lambda}{H}_{1}$. Here ${H}_{0}$ is the exactly solvable part and ${H}_{1}$ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order ${\ensuremath{\lambda}}^{2}$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.

DOI
10.1103/physrevb.92.165126

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Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ in the Hamiltonian H=H0+\ensuremathλH1H={H}_{0}+\ensuremath{\lambda}{H}_{1}H=H0​+\ensuremathλH1​. Here H0{H}_{0}H0​ is the exactly solvable part and H1{H}_{1}H1​ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order \ensuremathλ2{\ensuremath{\lambda}}^{2}\ensuremathλ2. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.
In a previous work [N. H. Tong, Phys. Rev. B 92 (2015) 165126], an equation-of-motion-based series expansion formalism was used to do the second-order strong-coupling expansion for the single-particle Green function of the Anderson impurity model (AIM). In this paper, we improve this theory in two aspects. We first use a more accurate scheme to self-consistently calculate the averages that appear in [Formula: see text]. In the resummation process, we use updated coefficients for the continued fraction (CF), guided by the formally exact CF from the Mori–Zwanzig theory. These changes lead to more accurate impurity spin responses to the magnetic bias of the bath. Combined with the dynamical mean-field theory, our theory gives improved description for the antiferromagnetism of Hubbard model at half-filling.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
We consider the cumulant expansion of the PAM employing the hybridization as perturbation (Phys. Rev. B 50, 17933 (1994)), and we obtain formally exact one-electron Green's functions (GF). These GF contain effective cumulants that are as difficult to calculate as the original GF, and the Atomic Approximation consists in substituting the effective cumulants by the ones that correspond to the atomic case, namely by taking a conduction band of zeroth width and local hybridization. This approximation has already been used for the case of infinite electronic repulsion U (Phys. Rev. B 62, 7882 (2000)), and here we extend the treatment to the case of finite U. The method can also be applied to the single impurity Anderson model (SIAM), and we give explicit expressions of the approximate GF both for the PAM and the SIAM.
We extend the self-consistent Green's functions formalism to take into account three-body interactions. We analyze the perturbative expansion in terms of Feynman diagrams and define effective one- and two-body interactions, which allows for a substantial reduction of the number of diagrams. The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fully correlated density matrices. We give examples up to third order in perturbation theory. To define nonperturbative approximations, we extend the equation-of-motion method in the presence of three-body interactions. We propose schemes that can provide nonperturbative resummation of three-body interactions. We also discuss two different extensions of the Koltun sum rule to compute the ground state of a many-body system.
The impurity Anderson model with a finite Coulomb repulsion U has been studied by using the conventional equation-of-motion treatment for the f-electron Green's function. To get a closed equation, two types of approximations are incorporated. One is to neglect, in a large U limit, higher order Green's functions which involve three f-electron annihilation operators. The other is to linearize the equation of motion by using a projection operator method. The average valency nf{\mathit{n}}_{\mathit{f}}nf​ and the spectral density of states (DOS) of f electrons are evaluated from the self-consistent equation for the Green's function. Results for nf{\mathit{n}}_{\mathit{f}}nf​ are in good agreement with exact results of Bethe ansatz method, and the obtained spectral DOS exhibits a well-known three-peak structure. Effects of finite U and finite temperature on the spectral DOS are discussed.
A new approach to many-body quasiparticle dynamics of the Anderson impurity model at flnite temperatures is proposed in the framework of the equation-of-motion method.We flnd a new exact identity, which relates the one-particle and many-particle Green functions.Using this identity, it is shown that a new interpolating approximation for the one-particle Green function can be constructed.The approximation which simultaneously reproduces tle weak-coupling limit and the strong coupling limit can be formulated self-consistently.This approach offers a new method for a systematic construction of the approximate interpolating dynamical solutions of strongly correlated electron models.