We present numerical renormalization group (NRG) calculations for a single-impurity Anderson model with a linear coupling to a local phonon mode. We calculate dynamical response functions, spectral densities, dynamic charge and spin susceptibilities. Being non-perturbative, the NRG is applicable for all parameter regimes. Our calculations cover both weak and strong electron-phonon coupling for zero and finite electron-electron interaction. We interpret the high-and low-energy features and compare our results to atomic limit calculations and perturbation theory. In certain restricted parameter regimes for strong electron-phonon coupling, a soft phonon mode develops inducing a very narrow resonance at the Fermi level.

We investigate electron-phonon coupling in many-electron systems using dynamical mean-field theory in combination with the numerical renormalization group. This non-perturbative method reveals significant precursor effects to the gap formation at intermediate coupling strengths. The emergence of a soft phonon mode and very strong lattice fluctuations can be understood in terms of Kondo-like physics due to the development of a double-well structure in the effective potential for the ions.Despite the many years of study of the electron-phonon interaction in metallic systems, there remain fundamental problems that have yet to be resolved; particularly in the strong-coupling regime and in conjunction with strong electron-electron interactions. A solution to these problems will be required to understand fully phenomena such as the colossal magnetoresistance effect in manganites [1]. Also in the metallic alkaline-doped C 60 -based compounds, high critical superconducting temperatures have been observed [2]. These materials are known to have strong electron-phonon and electron-electron interactions [2]. Recent photoemission experiments indicate strong electron-phonon coupling in the cuprate hightemperature superconductors [3]. There is a clear need of theoretical techniques to tackle these problems in the strong-coupling regime.Although electron-phonon problems involving one or few electrons can be solved to very high accuracy [4,5], so far there are no comparably accurate approaches for the many-electron case relevant to metallic systems. In this letter we study the simplest realization of electronphonon coupling: The Holstein model with finite electron density describes the coupling of Einstein (LO) phonons to the density of electrons of a non-degenerate conduction band:No general exact solution of this model is known for systems with finite electron density, even in the limit of infinite spatial dimensions (d = ∞). This limit takes local quantum fluctuations fully into account and has proved to be a powerful tool in understanding strongly correlated systems [6,7]. Although exactly solvable for d = ∞, the case of a single electron in the band [5] is physically very different from the many-electron case since no electron-electron pairing (bipolaron formation, superconductivity etc.) can occur. Another, more instructive limiting case is the static limit ω 0 = 0, where the phonons are replaced by a static displacement of the lattice ('static' or 'classical' approximation). Extensive calculations in this limit for d = ∞ have been presented in Ref. 8. However, it is immediately clear that this static limit neglects all possible effects stemming from the quantum nature of the lattice excitations. In the opposite limit of ω 0 → ∞ the lattice reacts instantaneously to the state of the electrons. Here, the Holstein model can be mapped onto a non-retarded attractive Hubbard model [9] by integrating out the phonons. The Hubbard model has been intensively studied, and much recent progress has been based on using the d = ∞ limit....

We give a comprehensive analysis of the singular dynamics and of the low-energy fixed point of one-channel impurity s-d models with ferromagnetic and underscreened antiferromagnetic couplings. We use the numerical renormalization group (NRG) to perform calculations at T = 0. The spectral densities of the one-electron Green's functions and t-matrices are found to have very sharp cusps at the Fermi level (ω = 0), but do not diverge. The approach of the Fermi level is governed by terms proportional to 1/ ln 2 (ω/T0) as ω → 0. The scaled NRG energy levels show a slow convergence as 1/(N + C) to their fixed point values, where N is the iteration number and C is a constant dependent on the coupling J from which the low energy scale T0 can be deduced. We calculate also the dynamical spin susceptibility, and the elastic and inelastic scattering cross-sections as a function of ω. The inelastic scattering goes to zero as ω → 0, as expected for a Fermi liquid, but anomalously slowly compared to the fully screened case. We obtain the asymptotic forms for the phase shifts for elastic scattering of the quasiparticles in the high-spin and low-spin channels.

Abstract. We show that the low energy behaviour of quite diverse impurity systems can be described by a single renormalized Anderson model, with three parameters, an effective levelǫ d , an effective hybridizatioñ V , and a quasiparticle interactionŨ . The renormalized parameters are calculated as a function of the bare parameters for a number of impurity models, including those with coupling to phonons and a FalikovKimball interaction term. In the model with a coupling to phonons we determine where the interaction of the quasiparticles changes sign as a function of the electron-phonon coupling. In the model with a FalikovKimball interaction we show that to a good approximation the low energy behaviour corresponds to that of a bare Anderson model with a shifted impurity level.PACS. 71.10.-w Theories and models of many-electron systems -71.27.+a Strongly correlated electron systems; heavy fermions -75.20.Hr Local moment in compounds and alloys; Kondo effect, valence fluctuations, heavy fermions

PACS. 71.10.Fd -Lattice fermion models (Hubbard model, etc.). PACS. 71.30.+h -Metal-insulator transitions and other electronic transitions . PACS. 71.38.-k -Polarons and electron-phonon interactions (see also 63.20.Kr Phonon-electron interactions in lattices).Abstract. -We investigate metal-insulator transitions in the half-filled Holstein-Hubbard model as a function of the on-site electron-electron interaction U and the electron-phonon coupling g. We use several different numerical methods to calculate the phase diagram, the results of which are in excellent agreement. When the electron-electron interaction U is dominant the transition is to a Mott-insulator; when the electron-phonon interaction dominates, the transition is to a localised bipolaronic state. In the former case, the transition is always found to be second order. This is in contrast to the transition to the bipolaronic state, which is clearly first order for larger values of U . We also present results for the quasiparticle weight and the double-occupancy as function of U and g.c EDP Sciences