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Physical review. B, Condensed matter

Improved treatment of retarded self-energy calculations in perturbation theory

S. Wermbter

Published Apr 15, 1996• 2 citations
Publisher

Abstract

I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.10569

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I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.
I present a numerical treatment for the retarded self-energy in two dimensions (2D), which is formally exact for the frequency and the momentum sums occurring in perturbation theory with respect to the Coulomb interaction U. In this frequency- and momentum-renormalization-group technique, which allows the sequential elimination of regions of high frequencies and momenta, I focus on the 2D fluctuation-exchange approximation for the Hubbard model. I present results for the self-energy and the spectral density down to temperatures t=${10}^{\mathrm{\ensuremath{-}}4}$/t (the next neighbor hopping amplitude). The obtained result shows evidence for a Fermi-liquid phase for bandfillings n near half filling for U/t=2 to U/t=8 for T\ensuremath{\rightarrow}0.
We present self-consistent Green-function calculations for the properties of the symmetric Anderson impurity model at zero temperature. A stable method for calculating the spectral density directly on the real frequency axis in the spin-fluctuation limit is described. Results using an approximation to the self-energy, previously termed the fluctuation-exchange approximation, which includes the effects of spin and density fluctuations (within the random-phase approximation) and multiple electron-electron scattering are compared with second-order perturbation-theory calculations. The static spin and charge susceptibilities, calculated in a manner consistent with conservation laws, are compared with the exact results from the Bethe ansatz. We find that the fluctuation-exchange approximation gives a good description of the charge susceptibility over a wide range of interaction strengths and is a great improvement over the self-consistent second-order theory. However, the T-linear coefficient of the specific heat is greatly overestimated by the fluctuation-exchange approximation and this leads to the underestimation of the spin susceptibility. The self-consistent second-order perturbation theory, which underestimates the specific-heat coefficient, predicts a magnetic ground state for sufficiently high interaction strengths.
We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.
A strictly truncated (weak-coupling) perturbation theory is applied to the attractive Holstein and Hubbard models in infinite dimensions. These results are qualified by comparison with essentially exact quantum Monte Carlo results. The second-order iterated perturbation theory is shown to be quite accurate in calculating transition temperatures for retarded interactions, but is not as accurate for the self-energy or the irreducible vertex functions themselves. Iterated perturbation theory is carried out through fourth order for the Hubbard model. The self-energy is quite accurately reproduced by the theory, but the vertex functions are not. Anomalous behavior occurs near half-filling because the iterated perturbation theory is not a conserving approximation.
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.