Percolation cluster numbers by real-space renormalization

The average numbern _s (p) of percolation clusters withs sites is calculated for the triangular lattice using real-space renormalization. Fors up to 2,000 the whole range of concentrationsp was analyzed;n _s varied over sixty decades. We found logn _s –s forp belowp _c =0.5, andn _s s ^–, =2.35 atp=p _c . For smallp one hasn _s (p · ) ^s . Nearp _c we found the scaling fromn _s s ^– f((p _c –p) ·s ) with=0.53. Presumably for the first time renormalization methods were used to calculate percolation properties not only nearp _c but also far away from the critical point.Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Correlated percolation — a real-space renormalization group study

A quenched-correlated percolation system is studied by using the real-space renormalization group method. We chose the nearest-neighbour short-range order α as the correlation parameter. By treating the correlated probability self-consistently in terms of site occupation probability p and α, we found that, in two dimensions, there is only one nontrivial physical fixed point at random percolation and the correlation is an irrelevant parameter which always leads to the same universality.     

Anderson localization in a periodic photonic lattice with a disordered boundary

We investigate experimentally the light evolution inside a two-dimensional finite periodic array of weakly coupled optical waveguides with a disordered boundary. For a completely localized initial condition away from the surface, we find that the disordered boundary induces an asymptotic localization in the bulk, centered around the initial position of the input beam.     

A theory for the Anderson lattice

Zeitschrift für Physik B Condensed Matter - A perturbational analysis of the Anderson lattice problem in the limit of infinite local Coulomb repulsion is given. It furnishes a clear conceptual...     

Differential real-space renormalization of the Gaussian model in three dimensions

We show how the differential real-space renormalization method can be extended beyond two dimensions by applying it to the Gaussian model on the fcc and diamond lattice.     

Study of Anderson localization by a continued-fraction method

The electron localization is studied for Anderson's tight-binding model of a disordered two-dimensional square lattice. For a large system of 10⁴ sites the averaged squared modulus $\text{| G}{}_{00}\text{|}{}^2$ of the Green's function is evaluated by a continued-fraction method. From this quantity, following Anderson's criterion, the energy of the mobility edge is found as a function of the degree of disorder. Also the Anderson transition is recognized.     

Destruction of Anderson localization by a weak nonlinearity

We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schr?dinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time proportional, variant t alpha, with the exponent alpha being in the range 0.3-0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.     

Iron pncitide superconductors studied by the functional renormalization group

We review the functional renormalization group (fRG) approach to the superconducting iron pnictides. We start with simple two-pocket models for the basic weak-coupling picture and then build up the complexity of the many-orbital problem in two steps. In this way we discuss what one can learn about the phase diagrams, superconducting pairing mechanism and competing orders in iron arsenides. Special attention is devoted to where this theoretical approach exposes similarities and differences between the physics of the pnictides and that of the high-T _c cuprates. Finally, we describe some challenges for getting a consistent theoretical understanding of the new iron superconductor material class in a combination of ab-initio and functional renormalization group approaches.     

The Schrieffer-Wolff transformation for the underscreened Anderson lattice

We present a derivation of the Schrieffer-Wolff transformation for the Anderson Lattice Hamiltonian with a two-fold degenerate f-level in each site. The degeneracy of the f-electrons has been taken into account in order to describe uranium and other actinide magnetic compounds with a spin larger than , for example a total S=1 spin for the f-electrons. The transformed Hamiltonian has several terms as in the classical case, but we have obtained here both an exchange (Kondo) interaction between the S=1 f-spins and the spins of the conduction electrons, and also an effective f-band term. This f-band term describes better the underscreened Kondo lattice model which has been recently developed to explain the Kondo-ferromagnetism coexistence observed in uranium compounds such as UTe [N.B. Perkins, M.D. Nunez-Regueiro, J. R. Iglesias, B. Coqblin, Phys. Rev. B 76 (2007) 125101].     

A renormalization group study of the Anderson Hamiltonian for arbitrary band-filling

The Anderson hamiltonian has been studied for arbitrarily filled bands using the real space renormalization group (RSRG) method. The ground state energy is determined in one dimension for the scale factors n_s = 3 and n_s = 5. Direct application of RSRG gives, for a certain n_s, the results for only n_s number of discrete cases of band-filling. We then devise an interpolation scheme which spans the entire region of the band and gives the ground state energy for any general filling. The results, as provided by our scheme, satisfy the desired particle-hole symmetry exactly and are always an upper bound to the exact one. The non-linear dependence of the band energy on its filling is represented correctly, even for small n_s. Finally, we discuss how the results improve for large n_s.     

Transport properties of the Anderson lattice

Transport properties including electrical resistivity, thermoelectric power, Lorenz number and a.c. conductivity are evaluated in an approximate fashion for the Anderson lattice model for six-fold degenerate Ce ions. Coherence (Bloch's theorem) is explicitly included while the effects of intersite interactions which may be responsible for magnetic and superconducting instabilities are neglected. The calculations utilize the AverageT-matrix Approximation (ATA) with the self-consistent Non-Crossing Approximation (NCA) perturbation theory employed to give the single siteT-matrix estimate. The resistivity peaks near the characteristic Kondo temperatureT ₀, with high temperature logarithmic decrease and low temperatureT ² behavior. The thermoelectric power is positive and similar to the impurity result except for low temperatures; sign changes in the thermopower are in principle possible with momentum dependent hybridization. Frequency and temperature dependent optical conductivity calculations are in qualitative agreement with experimental data, although a suitably defined optical effective mass and scattering rate do not agree at least for large orbital degeneracy. The behavior of these latter quantities is qualitatively different for twofold degeneracy. Unanswered questions arising from the experimental literature are summarized.     

Self-consistent theory of localization for the Anderson model

The self-consistent theory of electron localization in a random system in the form proposed by Vollhardt and Wölfle is generalized for the analysis of localization in the Anderson model. We derive the general equations appropriate for the system with rather general form of the electronic spectrum. Explicit calculations are restricted to the lattices of cubic symmetry and use the effective mass approximation to obtain the final results. Anderson's critical ratio for the localization of all the electronic states in the tight-binding band is evaluated and found to be in surprisingly good agreement with the results of numerical analysis of localization in the Anderson model.