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arXiv:1512.00173 · Annalen der Physik

Non‐interacting single‐impurity Anderson model: solution without using the equation‐of‐motion method

Zakaria M. M. Mahmoud, Florian Gebhard

Published Oct 16, 2015• 4 citations
arXivHTMLPDF

Abstract

Ground‐state properties of the non‐interacting symmetric single‐impurity Anderson model (SIAM) are derived from the corresponding eigenenergy equation. Explicit formulae are given for the ground‐state energy, the hybridization, and the momentum distribution that are essential quantities for variational approaches to the interacting model. Various spectral functions, e.g., the total density of states, the phase shift function, and the impurity spectral function, are shown to agree with those obtained from the equation‐of‐motion method (see supplementary material). For a constant hybridization strength and a semi‐elliptic host density of states it is seen that the impurity spectral function builds up weight at the band edges. image

DOI
10.1002/andp.201500259

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Ground‐state properties of the non‐interacting symmetric single‐impurity Anderson model (SIAM) are derived from the corresponding eigenenergy equation. Explicit formulae are given for the ground‐state energy, the hybridization, and the momentum distribution that are essential quantities for variational approaches to the interacting model. Various spectral functions, e.g., the total density of states, the phase shift function, and the impurity spectral function, are shown to agree with those obtained from the equation‐of‐motion method (see supplementary material). For a constant hybridization strength and a semi‐elliptic host density of states it is seen that the impurity spectral function builds up weight at the band edges. image
A dynamic density-matrix renormalization group approach to the spectral properties of quantum impurity problems is presented. The method is demonstrated on the spectral density of the flat-band symmetric single-impurity Anderson model. We show that this approach provides the impurity spectral density for all frequencies and coupling strengths. In particular, Hubbard satellites at high energy can be obtained with a good resolution. The main difficulties are the necessary discretization of the host band hybridized with the impurity and the resolution of sharp spectral features such as the Abrikosov–Suhl resonance.
Single impurity Anderson model describes a system consisting of non-interacting conduction electrons coupled with a localized orbital having strongly interacting electrons at a particular site. This model has been proven successful to explain the phenomenon of metal-insulator transition through Anderson localization. Despite the well-understood behaviors of the model, little has been explored theoretically on how the model properties gradually evolve as functions of hybridization parameter, interaction energy, impurity concentration, and temperature. Here, we propose to do a theoretical study on those aspects of a single impurity Anderson model using the distributional exact diagonalization method. We solve the model Hamiltonian by randomly generating sampling distribution of some conducting electron energy levels with various number of occupying electrons. The resulting eigenvalues and eigenstates are then used to define the local single-particle Green function for each sampled electron energy distribution using Lehmann representation. Later, we extract the corresponding self-energy of each distribution, then average over all the distributions and construct the local Green function of the system to calculate the density of states. We repeat this procedure for various values of those controllable parameters, and discuss our results in connection with the criteria of the occurrence of metal-insulator transition in this system.
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 analyze the ground-state energy, magnetization, magnetic susceptibility, and Kondo screening cloud of the symmetric single-impurity Anderson model (SIAM) that is characterized by the bandwidth WWW, the impurity interaction strength UUU, and the local hybridization VVV. We compare Gutzwiller variational and magnetic Hartree-Fock results in the thermodynamic limit with numerically exact data from the density-matrix renormalization group (DMRG) method on large rings. To improve the DMRG performance, we use a canonical transformation to map the SIAM onto a chain with half the system size and open boundary conditions. We compare to Bethe ansatz results for the ground-state energy, magnetization, and spin susceptibility that become exact in the wide-band limit. Our detailed comparison shows that the field-theoretical description is applicable to the SIAM on a ring for a broad parameter range. Hartree-Fock theory gives an excellent ground-state energy and local moment for intermediate and strong interactions. However, it lacks spin fluctuations and thus cannot screen the impurity spin. The Gutzwiller variational energy bound becomes very poor for large interactions because it does not describe properly the charge fluctuations. Nevertheless, the Gutzwiller approach provides a qualitatively correct description of the zero-field susceptibility and the Kondo screening cloud. The DMRG provides excellent data for the ground-state energy and the magnetization for finite external fields. At strong interactions, finite-size effects make it extremely difficult to recover the exponentially large zero-field susceptibility and the mesoscopically large Kondo screening cloud.
The ground-state energy, wave-function, hybridization gap and magnetic phasediagram of the Single Site Impurity Anderson Model (SIAM) and the Periodic AndersonModel (PAM) were calculated using Exact-Diagonalization (ED) technique. The EDtechnique was applied to the SIAM and PAM in one dimension before speculating on Ce 4fbeing a test bed of the results, since it has 4f localized electrons in the Heavy fermioncompound CeCu2Si2. Each of the results obtained describe the evolution of the electronicsystem in terms of a simple set of parameters.The parameter regimes studied suggest asmooth phase transition from an antiferromagnetic (AFM) phase to a ferromagnetic (FM)phase at quarter fillings in one dimension when t increases and a magnetic instability wasobserved as U → ∞. The role of hybridization between the conduction electrons and thelocalized f orbital in the Anderson impurity in a semiconductor was also investigated as Vincreases in the SIAM and the PAM.
We study the non‐interacting single‐impurity Anderson model (resonant level model) on a lattice at finite temperature as an illustrative example for an exactly solvable quantum‐mechanical problem, and derive the free energy, various thermodynamic potentials (internal energy, entropy, magnetization), and response functions (specific heat, zero‐field magnetic susceptibility). We calculate the magnetic screening cloud, and derive the corresponding correlation length in one dimension beyond which the correlations decay exponentially. The present results remain qualitatively applicable for the interacting single‐impurity Anderson model when the energy scale Г of the resonant‐level model is replaced by the Kondo scale TK.