Anderson impurity model

In the case of a single site junction with two (spin-up and spin-down) states and Coulomb interaction between these states (Anderson impurity model), the linear conductance properties have been successfully studied by means of the EOM approach in the cases related to Coulomb blockade [203, 204] and the Kondo effect [205]. Later the same method was applied to some two-site models [206-209], Multi-level systems were started to be considered only recently [210,211], Besides, there are some difficulties in building the lesser GF in the nonequilibrium case (at finite bias voltages) by means of the EOM method [212-214],... 

In the limit of a single-level quantum dot (which is, however, a two-level system because of spin degeneration) we get the Anderson impurity model (AIM)... 

We consider the following model Hamiltonian (which can be called the multilevel Anderson impurity model, the Hubbard model, or the quantum cluster model)... 

This model is quite universal, describing a variety of correlated electron systems coupled to the leads the Anderson impurity model, the multilevel quantum dot with diagonal noninteracting Hamiltonian quantum dots, when the off-diagonal matrix elements of eap describe hopping between individual dots, and, finally, the ID and 2D quantum point contacts. 

The Anderson impurity model is used to describe the Coulomb interaction on a single site ... 

XANES is one of powerful tools for the study of chemical states of lanthanoid compounds. Most of the conclusions drawn from the XANES for the lanthanoid compounds have been concerned with the chemical states or characteristics of the electronic structures. In particular, valences of lanthanoids have been studied using such spectra (2). They were obtained by assigning some of the peaks in the spectra to different valences. By using the Anderson impurity model, the valences were derived from intensities of shake-up peaks (3). 

Bianconi ct al. adopted the Anderson impurity model and assumed prominent change of the electronic structure which resulted in the shake-up (3). This restructuring of the bands was attributed to a large 4/ - core hole interaction. Soldatov et al. took the same shake-up model and successfully got theoretical XANES with the peaks A-D, by convoluting multiplescattering XANES spectra obtained for the two electron configurations for the final states due to the shake-up (6). 

The intensity of the weaker structure is observed to depend on the partner element in the intermetallic. The manifestation of intermediate valence in the 3d — 4/ XAS and other core and outer spectra of Ce intermetallics has been interpreted in the framework of the Anderson impurity model [628]. 

The results this section are very relevant to the discussion in chapter 11 of the Anderson impurity model and the quasiatomic orbital collapse model. It is significant, in particular, that the transition occurs for rather small clusters, smaller than might perhaps be expected for the impurity model to be applicable. The quasiatomic model provides a straightforward explanation for the varying degrees of oxidation observed in [695] the chemical activity of the lanthanide atoms is greater when the orbitals are in an expanded or outer-well state than when they are in a contracted or inner-well one. On the other hand, an important issue which needs to be determined is over what range of cluster sizes an effective conduction band actually appears, since its presence provides the hybridisation forces which play a crucial role in the impurity model. 

Whilst there have been several theoretical investigations of the effect of hybridisation on the crystal-field excitations within the ground multiplet (Maekawa et al. 1985, Lopes and Coqblin 1986), there have been relatively few in which the spin-orbit level is explicitly included. Cox et al. (1986) have shown, in the context of the Anderson impurity model, that when is comparable to the spin-orbit splitting, the inelastic peak is broadened and shifted to lower energies. Given that the cross-section is weak, at about half the intensity of the praseodymium spin-orbit cross-section, they concluded that the transition was unlikely to be seen except in heavy-fermion compounds with low values of This appears to be confirmed by the failure to observe such a transition in CePdj in recent measurements on HET (Osborn, unpublished). On the other hand, the... 

The computational technique used to treat the generalized Anderson impurity model in the slave boson representation will be described in some detail. For an extensive discussion see Coleman (1984). In Appendix A we represented the CEF states of stable 4f" shells (i.e., with integer occupation n) by pseudofermions. In the present case of unstable shells with possible 4f and 4f configurations we need an additional slave boson field for the 4f° state. The interesting physical quantities, static as well as dynamic, can be calculated in terms of the fully renormalized fermion and slave boson Matsubara Green s functions... 

This static alloy analogy picture should be a good description as long as there is a separation of timescales. If this breaks down, dynamical fluctuations—or quantum fluctuations— which are beyond this static picture, become important. These quantum fluctuations are the main emphasis of DMFT (Georges et al., 1996), which maps the system onto an effective Anderson impurity model, describing a dynamically fluctuating impurity in a self-consistently determined effective host. So far, DMFT has been formulated for model Hamiltonians, such as the Hubbard model, and material-specific results have been achieved by constructing these model Hamiltonians from realistic band structure calculations. In... 

Since the model (1) is very difficult to solve, most work has been limited to the Anderson impurity model, where only the f-level on one site is considered. This is perhaps the simplest model of a mixed valence system, since the model only contains the hopping term necessary to couple different f-conligurations and the important f-f Coulomb interaction. Variations of this model have been very widely used to describe transport, thermodynamic and spectroscopic properties of mixed valence... 

To end up the discussion of the attempts to calculate spectral functions for the Anderson impurity model we shortly mention decoupling schemes for higher-order Green s functions (Hewson 1966, Theumann 1969, Lacroix 1981, Oh and Doniach 1982, Czycholl 1985). The approximations in this method consist of replacements of the type )) Apart from the fact that the... 

In this chapter different methods for calculating electron spectra of the Anderson impurity model have been presented. In particular, the recent development in terms of /N expansions was emphasized. It was demonstrated that for T = 0 there are now fairly accurate calculations available, even for small values of the degeneracy Nf. In particular it is possible to obtain a fair description of the so-called Kondo resonance. 

The extension of the work described here to the Anderson lattice is an interesting but difficult problem. It would also be interesting to include more terms in the Anderson impurity model, describing for instance multiplet effects. Auger decay and the direct f-d Coulomb interaction. The determination of the parameters in the model from first principles is an important problem. Much progress has been made by Herbst et al. (1978) and by Herbst and Wilkins (1979) for 17, Ui and % and there are promising attempts to estimate from band structure calculations. 

The spectra of the Anderson impurity model have been compared extensively with experimental results for many different lanthanide compounds. Some of this work was reviewed in section 5. It was shown that the model gives a surprisingly good description of the experimental results. In particular, it was shown that for a given compound essentially the same parameters can be used to describe the different spectroscopies. It was, however, also found that the neglected multiplet effects often are important and that there are broadening effects, which are not included in the model. 

Gunnarsson and Schonhammer (1983a,b,c, 1985a,b see also chapter 64 in this volume) proposed a description of Ce spectroscopic data in terms of an Anderson impurity model, which allows to extract the crucial parameters, i.e., the 4f conduction band hybridization A and the 4f occupancy from spectroscopic data. The justification for using a single-impurity Anderson Hamiltonian comes from spectroscopic data like the BIS data discussed here or photoemission and absorption data discussed in other chapters of this book. The fact that A increases, e.g., for Ce-Ni... 

Fig. 30. Intensity of f final state in BIS, normalized to total f intensity, over f° contribution in the ground state, derived from spectra calculated with the Anderson impurity model (Gunnarsson and Schonhammer 1983a,b).

The f-count determination of ICF compounds at the beginning of the rare earth series in CEELS is therefore hampered by final-state effects as in other core level spectroscopies. In comparison to XAS CEELS shows further complications as a result of more complicated final-state multiplet structure (Strasser et al. 1983b, Hillebrecht et al. 1986). However, Schneider et al. (1985) have included recently analysis of 3d CEELS of La and Ce compounds with fractional f occupation into the many-body formalism of an Anderson impurity model. The agreement of relative energy positions between calculated and measured spectra was encouraging, but in order to assess fully the consequences of initial-state mixing in the final state, multiplet splitting has to be included into the many-body formalism. 

Fig. 35. Predictions of a slave-boson treatment of a Kondo insulator for the specific heat Cp (solid line) and temperature derivative of the f-occupation number dUf/dT (solid circles). As opposed to the Anderson impurity model, where dnf/dT peaks at a temperature which is 2-3 times larger than the temperature where Cp is maximum, for this Anderson Lattice calculation both quantities peak at roughly the same temperature. From Riseborough (1992).

Despite the qualitative agreement, there appear to be discrepancies and additional structures that do not seem to be attributable to contaminants or surface shifts. These anomalous features become more prominent for the systems that are expected to have the narrowest f bands and are most extreme for the heavy fermion class of uranium systems. In view of the large value of the imderlying non-interacting f band width A, it therefore seems reasonable to assume that the effect of Coulomb interactions may be introduced via perturbation theory. Thus, within the formalism of the Anderson impurity model, or even the Hubbard model, the spectra bear resemblance to a Lorentzian band of width A 2 e ( and since the 14-fold degenerate f band is expected to contain only 2 or 3 electrons, most of the spectral wei t is located above the Fermi level fi. 

Encouraged by the correct description of the thermodynamic properties, of the electron-hole symmetric Anderson impurity model (Horvatic and Zlatic 1985, Okada et al. 1987), by straightforward perturbation theory in Us, several groups have examined the spectral density of the Anderson lattice model using second-order perturbation theory... 

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