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physica status solidi (b)

Ground state of the impurity Anderson model revisited: A projector operator solution

P. Roura-Bas, Ignacio J. Hamad, E. V. Anda

Published Dec 10, 2014• 5 citations
Publisher

Abstract

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.

DOI
10.1002/pssb.201451520

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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.
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.
The Kondo model describes a magnetic impurity in a metallic host. Here, the authors use variational approaches (Lanczos, Yosida, Gutzwiller) and quasi-exact numerical techniques (numerical renormalization group, density matrix renormalization group) to study the ground-state energy and magnetization, and compare these properties to Bethe ansatz results. The Gutzwiller wave function is an excellent trial state for strong coupling. For weak coupling, the numerical renormalization group gives the best description of the magnetic properties. A combination of Bethe ansatz and perturbation theory provides new benchmark results for the zero-field susceptibility.
We consider an integrable model describing an Anderson-like impurity coupled to an open ttt-JJJ chain. Both the hybridization (i.e., its coupling to bulk chain) and the local spectrum can be controlled without breaking the integrability of the model. As the hybridization is varied, holon and spinon bound states appear in the many body ground state. Based on the exact solution we study the state of the impurity and its contribution to thermodynamic quantities as a function of an applied magnetic field. Kondo behavior in the magnetic response of the impurity can be observed provided that its parameters have been adjusted properly to the energy scales of the holon and spinon excitations of the one-dimensional bulk.
Single-particle dynamics of the Anderson impurity model in the presence of a magnetic field H are considered, using a recently developed local moment approach that encompasses all energy scales, field and interaction strengths. For strong coupling in particular, the Kondo scaling regime is recovered. Here the frequency (ω/ω K ) and field ( H /ω K ) dependence of the resultant universal scaling spectrum is obtained in large part analytically, and the field-induced destruction of the Kondo resonance investigated. The scaling spectrum is found to exhibit the slow logarithmic tails recently shown to dominate the zero-field scaling spectrum. At the opposite extreme of the Fermi level, it gives asymptotically exact agreement with results for statics known from the Bethe ansatz. Good agreement is also found with the frequency and field-dependence of recent numerical renormalization group calculations. Differential conductance experiments on quantum dots in the presence of a magnetic field are likewise considered, and appear to be well accounted for by the theory. Some new exact results for the problem are also established.
We analyze the two-Anderson-impurity problem, in the strong-Coulomb-repulsion limit, by means of variational wave functions whose equations we solve analytically. We found two pairs of doublet states, one odd and one even with respect to the midplane between the impurities. These doublets make a significatively strong use of the hybridization terms of the Hamiltonian and have a much larger Kondo-like correlation energy than a single impurity (the Kondo energy). Furthermore, these doublets combine to form a supersinglet. This supersinglet makes use of the remaining hybridization probability to improve its energy with respect to the doublet energy, but for low-dimensional systems and at the interimpurity distances where the doublets are fully developed, its energy gain is exponentially small. The interaction behind the doublets is a first-order (in the effective Kondo coupling) one; it also generates a parallel alignment of the impurity spins. This ferromagnetic impurity-impurity response is thus generated without resort to the RKKY interaction. The effective range of this first-order Doublet interaction is also much larger than the one of the RKKY interaction.
The Anderson Impurity Model is a generic model to describe systems in which one or more energy levels are coupled to an electronic host. On the energy levels the electrons are subject to a Coulomb interaction. Possible applications of this model are magnetic atoms in a metal or quantum dot systems. Already for one impurity it displays very interesting physics including the well known Kondo effect. For two and more impurities the properties of the system in the local moment regime are determined by an interplay of the Kondo effect and the coupling between the impurity spins. Due to the interplay of multiple energy scales, this model poses a serious task for any computational tool. For a single impurity the model can be very efficiently treated using Wilson"s numerical renormalization group (NRG). Since the numerical effort of NRG calculations rises exponentially with the number of impurities, generically it cannot be applied to systems with more than two impurities. The functional renormalization group (fRG) has been very successfully applied to quantum impurity problems in the past. In this work it will be used to calculate the single-particle dynamics for Anderson-like quantum impurity systems in equilibrium. A comparison to NRG in the case of a single impurity will serve as a benchmark before systems with two impurities and quantum dot systems with two and three dots will be studied. The interplay between the Kondo effect and magnetic and orbital ordering will be investigated in these systems. Although the truncation in the presented fRG scheme is motivated by second order perturbation theory, it will be shown that fRG works also in those cases where perturbation theory fails. Generally it will turn out that fRG is capable of treating a broad variety of Anderson impurity systems within the same framework, including asymmetric impurity levels and interaction between multiple impurities.