If one wants to deal with strongly correlated electron lattice systems within dynamical mean field theory, one needs an impurity solver, i.e. a solution for the dynamics of the Anderson magnetic impurity problem. The most ambitious approach to that problem has been the consistent t-matrix approximation (CTMA). The basic integral equations could be solved with a slow variable approximation, which compared well with a direct numerical solution. Fermi-liquid behavior was not obtained, presumably the normalscattering potential was treated approximately only. We show, how the normal scattering potential can be included exactly. (c) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

For a strongly correlated electron lattice, if treated within dynamical mean-field theory, the most advanced analytical approach for the infinite U Anderson model as impurity solver is the consistent T-matrix approximation (CTMA). Recently [H. Keiter, K. Baumgartner, Phys. Staus Solidi B 242 (2005) 377.] we solved the central integral equations of the CTMA numerically with self-adaptive nets, and analytically with a new version of the slow variable approximation. Both methods were in very good agreement. Further progress on that problem is reported.

Based on the ranks of reduced density matrices, we derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement. In a similar way we obtain necessary conditions for the separability of a given mixed state with respect to partitions of all particles of the system into subsets. The special case of pure states is discussed separately.

The basic integral equations for the consistent t-matrix approximation (CTMA) of the infinite U Anderson model are formally solved within the slow variable approximation. The solution is compared with a numerical C++ analysis using self adapting nets. Very good agreement is found. (c) 2005 WILEY-VCH Verlag GmbH T Co. KGaA, Weinheim.

Local and non-local generalizations of dynamical mean-field theory (DMFT) are presented. They come from a self-avoiding walk in a hypercubic lattice, in which the principle of exclusion and inclusion is used. The coherent potential approximation (CPA) can be treated in a similar way. CPA and DMFT can be mapped onto each other. The numerical corrections to the DMFT are relatively weak, with the non-crossing approximation (NCA) used for the single impurity part. NCA-generalizations similar to those of the Wolfle group are set up with direct self-consistent perturbation theory.

The dynamical mean-field theory (DMFT) maps the periodic Anderson-model for infinite U onto a single impurity model with an effective band. This effective band is approximately described by a self-avoiding loop through the lattice with a local scattering matrix and can be written as a power series in that matrix times the unperturbed nearest neighbor propagator. We show that the power series has a finite radius of convergence, which is given by the inverse of the connective constant. In the limit of infinite spatial dimensions, in which the DMFT becomes exact, the underlying self-avoiding loop has zero radius of convergence. We also further comment on the breakdown of one self-consistency condition in the DMFT.

Many-particle perturbation theory is dominated by the Rayleigh–Schrödinger expansion, because at fixed particle density each term of the power series in the strength of the interaction has the right dependence on the particle number N. The first self-consistent generalization, the Brillouin–Wigner expansion, however, is non-extensive already in 2nd order. In Feenberg's generalization the same feature is found within iterative perturbation theory, but the series can be rearranged in such a way that lower orders are extensive. Applications are suggested.

We use the so-called lace expansion of mathematical random walk theory to derive several generalizations of dynamical mean-field theory (DMFT) for strongly correlated electron systems (SCES). DMFT is based on a self-consistence condition, by which a lattice Hamiltonian for SCES is mapped onto the corresponding impurity problem. It is a local theory with a density of states of the impurity problem following from the local band Green function. So in the Fourier-transformed version the self-energy is independent of momenta. From a mathematical point of view, the self-consistence condition follows from a physical self-avoiding loop. We discuss non-local as well as local corrections and examine them numerically for the Anderson lattice in the large U limit. All calculations are performed at finite spatial dimensions.

In dynamical mean-field theory (DMFT) the Anderson lattice model is mapped onto the impurity model, with the density of states determined from a self-consistence condition (scc). The mapping is rigorous in infinite spatial dimensions d. It can be diagrammatically modelled as self-avoiding loops. While at finite d > 4 the number of mathematical self-avoiding loops is negligible compared to all random loops, to the mathematical scc at infinite d they contribute a fraction 1/e to all random loops. The limits of d → ∞ and infinite loop length cannot be interchanged, thus making 1/d-corrections to the scc questionable. We find the analogous result for the DMFT loop. We also discuss numerical difficulties arising in the infinite-U limit of the Anderson lattice model, and analytically simulate them.

We report on a scalable implementation of the configuration-selecting multireference configuration interaction method for massively parallel architectures with distributed memory. Based on a residue driven evaluation of the matrix elements, this approach allows the routine treatment of Hilbert spaces of well over 10(9) determinants, as well as the selective treatment of triple and quadruple excitations with respect to the reference space. We demonstrate the scalability of the method for up to 128 nodes on the IBM-SP2 and for up to 256 nodes on the Cray-T3E. We elaborate on the specific adaptation of the transition residue-based matrix element evaluation scheme that ensures the scalability and load balancing of the method. (C) 1999 John Wiley & Sons, Inc.

The infinite U limit of the Anderson lattice model can be formulated as a non-Hermitean Hamiltonian, where the usual hybridization is contained in the unperturbed part. The grand partition function is written as a functional integral, from which one obtains a perturbation expansion which automatically contains a linked diagram expansion. Further typical approximations to the functional integral are discussed.

The scale of relaxation times in glasses has led to generalizations of the Drude model of the dielectric function in terms of an integral, containing a Drude kernel and a probability distribution. This integral equation is solved by a Mellin or a Stieltjes transform. Beyond known results, we obtain the probability distribution of the Havriliak-Negami dielectric function. Even more general classes of dielectric models can be dealt with, using Mellin's transform. They may serve as checks for numerical procedures applied to the underlying ill-posed problem, if experimental data for the dielectric function are used.

We suggest that Pauli-principle violations, which automatically occur within approximations in many body calculations for strongly correlated electron systems, are the cause for well known pathologies at low temperatures. As an example, the (1/N) expansion, where N denotes the degree of degeneracy of the f-level in Anderson's impurity model, is carried out without Pauli-principle violations. The preliminary result shows that the energy shifts due to the hybridization, which are characteristic for slave boson mean-field theories, are contained in the present approach.

The Hall effect which includes an extraordinary Hall effect due to the skew scattering for strongly correlated Fermi systems has been investigated using an extended single impurity infinite-U Anderson Hamiltonian at low temperature. A first-order field approximation was performed to calculate the scattering T matrix which is associated with the f Green's function and used for all the temperature ranges in the non-crossing approximation in a small applied magnetic field B. The numerical results are in qualitative agreement with the experimental results which account for the main features of the Hall effect in heavy-fermion compounds.

Investigating the breakdown of the self-consistence condition, by which the Anderson lattice model in infinite spatial dimensions is mapped onto the single impurity model (SI), in the limit U → ∞ we revisited the non-crossing approximation (NCA) at finite magnetic field. As expected, at temperatures T well above the zero field single-impurity Kondo-temperature TK, the lattice and the impurity results are in perfect agreement. For T ⪡ TK we obtained only impurity results, into which we included skew scattering. The latter influences the zero field Hall coefficient, but the electrical resistivity and the thermal conductivity only in regions outside of the validity of the NCA, and leave the thermopower and the Lorenz ratio unchanged.

Brillouin-Wigner (BW) perturbation formulae can be rearranged into a form first proposed by Feenberg. Feenberg's perturbation formulae also follow from a variational principle. They are successfully tested at two typical problems, for which ordinary perturbation techniques completely fail. The first is to find conditions for the bosonization of the Tomonaga model perturbatively. The second is to clarify whether non-Fermi liquid behavior of the momentum distribution function of the Luttinger model can be achieved perturbatively.

In infinite spatial dimensions d, the Anderson lattice model can be maped onto the single impurity model (SI) with an effective density of states determined from a self-consistence loop. If the non-crossing approximation (NCA) is used for the SI model, this approach allows for testing known extensions of the NCA to the lattice, like the XNCA or the LNCA. The d → ∞ loop is shown to be identical with that obtained in the XNCA, but differs from that of the LNCA. The interrelations are presented both analytically and numerically. Two other approaches to overcome the problem over-renormalization, which occurs in the XNCA below the SI-Kondo temperature, are presented, and the first steps towards a d−1 expansion are sketched. Transport properties are calculated for different approximationsto the lattice and are partly unsatisfactory if compared to those measured in Ce-compounds. A positive Hall coefficient is calculated without using a skew-scattering mechanism.

Combining self-consistent perturbation theory with conventional renormalization group equations, a new systematic approach to Kondo-type models is developed, which yields analytical expressions for universal quantities like pseudofermion exponents and large effective couplings, usually out of reach from perturbative calculations. The new approach is related to the parquet-treatment of the x-ray-edge-singularity by Noziéres and coworkers in the late sixties and makes no use of a linearization of the band-electron-dispersion-relation, which is crucial for the Bethe Ansatz, conformal field theory, or numerical renormalization group treatment of the models. In addition it covers 1/N2-corrections to the well-known Non-Crossing-Approximation.

We present various approaches to the Anderson lattice model, starting from the limit of infinite spatial dimensions d. In this limit the lattice model can be mapped onto the single impurity model (SI) with an effective density of states determined from a selfconsistence condition (SCC). Similar approaches to the lattice in 3d were based on the non crossing approximation (NCA) for the SI, but differed in the SCC used. Relations between the different SCCs and new ones are given, results for transport quantities are discussed. A diagrammatical derivation of the SCC in infinite d is improved by including d−1corrections.

A systematic approach, which combines renormalization group equations and self-consistent perturbation theory for the dynamical properties of single impurity exchange models (which hitherto were obtained from numerical results or from approximations), analytically furnishes universal quantities like pseudofermion exponents and large effective couplings, depending on the symmetry of the underlying model.