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Self-consistent hybridization expansions for static properties of the Anderson impurity model

Ignacio J. Hamad, Roura Bas, Pablo Gines, Armando A. Aligia, E. V. Anda

Published Nov 1, 2015• 5 citations
OpenAlex

Abstract

Building an effective Hamiltonian utilizing a projector-operatorprocedure, we derive an approximation based on a selfconsistenthybridization expansion to study the ground stateproperties of the Anderson Impurity model.We applied the approximationto the general case of finite Coulomb repulsionU, extending previous work with the same formalism in theinfinite-U case. The ground state energy and their related zerotemperature properties are accurately obtained in the case inwhich U is large enough, but still finite, as compared with the rest of energy scales involved in the model. The results for thevalence of the impurity are compared with exact results thatwe obtain from equations derived using the Bethe ansatz andwith a perturbative approach. The magnetization and magneticsusceptibility is also compared with Bethe ansatz results. Inorder to do this comparison, we also show how to regularizethe Bethe ansatz integral equations necessary to calculate theimpurity valence, for arbitrary values of the parameters.

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Building an effective Hamiltonian utilizing a projector-operatorprocedure, we derive an approximation based on a selfconsistenthybridization expansion to study the ground stateproperties of the Anderson Impurity model.We applied the approximationto the general case of finite Coulomb repulsionU, extending previous work with the same formalism in theinfinite-U case. The ground state energy and their related zerotemperature properties are accurately obtained in the case inwhich U is large enough, but still finite, as compared with the rest of energy scales involved in the model. The results for thevalence of the impurity are compared with exact results thatwe obtain from equations derived using the Bethe ansatz andwith a perturbative approach. The magnetization and magneticsusceptibility is also compared with Bethe ansatz results. Inorder to do this comparison, we also show how to regularizethe Bethe ansatz integral equations necessary to calculate theimpurity valence, for arbitrary values of the parameters.
By means of a projector operator formalism we study the ground state properties of the Anderson Impurity Hamiltonian. The non-perturbative treatment of the model agrees with the previous one, obtained by Inagaki [Prog. Theor. Phys. 62, 1441 (1979)] by means of a perturbation expansion with respect the hybridization term. We go beyond the Inagaki's formalism to the next leading order. It provides a very accurate calculation of the energy spectrum in the total spin sector and, in particular, the ground state energy in the whole parameter space. For a one body spinless system, the dependence of the ground state energy as a function of the impurity level obtained by this procedure remarkably agrees with analytical results. For the many body case the occupancy of the impurity as a function of the parameters is studied and it agrees with the corresponding one obtained by using the Bethe ansatz and the Numerical Renormalization Group solution of the Hamiltonian. The magnetization and susceptibility of the impurity is analyzed by studying the response of the system to an external magnetic field, from which it is possible to extract the Kondo temperature. The dependence of the Kondo scale with the parameters of the model is in excellent agreement with well-known results.
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.
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.
We discuss the ground-state energy of the Anderson degenerate impurity model for both finite and infinite on-site Coulomb repulsion U. The goal of the paper is to demonstrate the applicability of a theoretical approach to static ground-state properties introduced recently. This approach is based on the projection technique and on the introduction of cumulants. It allows for proper treatment of strong electronic correlations that are incorporated in the model. It also guarantees the size consistency of the evaluated physical quantities. We start from a compact expression for the ground-state energy, which is expanded into a continued fraction. By summing up the infrared divergencies, a self-consistent equation for the ground-state energy is found. For degeneracy Nf{\mathit{N}}_{\mathit{f}}Nf​\ensuremath{\rightarrow}\ensuremath{\infty} and U\ensuremath{\rightarrow}\ensuremath{\infty} the equation can be solved rigorously and gives the well-known Varma-Yafet solution. For Nf{\mathit{N}}_{\mathit{f}}Nf​ and U both finite our procedure leads to the result of the variational approach of Gunnarsson and Sch\"onhammer.