In the beginning of the 1970's, Wilson developed the concept of a fully non-perturbative renormalization group transformation. Applied to the Kondo problem, this numerical renormalization group method (NRG) gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG has been later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method including some guidelines of how to calculate physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean field theory.

We describe the generalization of Wilson's Numerical Renormalization Group method to quantum impurity models with a bosonic bath, providing a general non-perturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, J(ω) ∝ ω s . We find clear evidence for a line of continuous quantum phase transitions for subohmic bath exponents 0 < s < 1; the line terminates in the well-known KosterlitzThouless transition at s = 1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.The Numerical Renormalization Group method (NRG) developed by Wilson [1] is a powerful tool for the investigation of the Kondo model and its generalizations [1,2,3,4]. In these models, a (possibly complex) impurity, such as a localized spin, couples to a fermionic bath. In the case of a spin-1 2 impurity coupled antiferromagnetically to a metallic bath, the impurity spin is screened below a characteristic scale T K , the Kondo temperature [3]. The strength of the NRG lies in its nonperturbative nature and the ability to resolve arbitrarily small energies [1]. A variety of thermodynamic and dynamic quantities can be calculated for a large number of impurity models in the whole parameter space [4,5].There is, however, a very important class of models for which the NRG method has not yet been developed: models with a coupling of the impurity to a bosonic bath [6]. The intensively studied spin-boson model [7,8] belongs to this class; its Hamiltonian is given byHere the Pauli-matrices σ j describe a spin, i.e., a generic two-level system, which is linearly coupled to a bath of harmonic oscillators, with creation (annihilation) operators a † i (a i ). The bare tunneling amplitude between the two spin states | ↑ and | ↓ is given by ∆, and ǫ is an additional bias. The ω i are the oscillator frequencies and λ i the coupling strengths between the oscillators and the local spin. The coupling between spin and bosonic bath is completely specified by the bath spectral functionOf particular interest are power-law spectrawhere the dimensionless parameter α characterizes the dissipation strength, and ω c is a cutoff energy. The value s = 1 corresponds to the case of ohmic dissipation.The spin-boson model is a generic model describing quantum dissipation; it has been discussed in the context of a great variety of physical problems [7,8] ranging from the effect of friction on the electron transfer in biomolecules [9] to the description of the quantum entanglement between a qubit and its environment [10,11,12].Considering the wealth of applications, the question arises whether Wilson's NRG method can be exploited for this class of models; and it is the purpose of this paper to show that this is indeed the case. What we have in mind here is the direct mapping of models like (1) to a semi-infinite chain form typical for ...

The zero-temperature transition from a paramagnetic metal to a paramagnetic insulator is investigated in the dynamical mean field theory for the Hubbard model. The self-energy of the effective impurity Anderson model (on which the Hubbard model is mapped ) is calculated using Wilson's numerical renormalization group method. Results for the quasiparticle weight, the spectral function, and the self-energy are discussed for the Bethe and the hypercubic lattice. In both cases, the metal-insulator transition is found to occur via the vanishing of a quasiparticle resonance that appears to be isolated from the Hubbard bands. PACS numbers: 71.10.Fd, 71.27. + a, 71.30. + h The Mott-Hubbard metal-insulator transition [1,2] is one of the most fascinating phenomena of strongly correlated electron systems. This transition from a paramagnetic metal to a paramagnetic insulator is found in various transition metal oxides, such as V 2 O 3 doped with Cr [3]. The mechanism driving the Mott-Hubbard transition is believed to be the local Coulomb repulsion U between electrons on a same lattice site, although the details of the transition should also be influenced by lattice degrees of freedom. Therefore, the simplest model to investigate the correlation driven metal-insulator transition is the Hubbard model(1) where c y is ͑c is ͒ denote creation (annihilation) operators for a fermion on site i, t is the hopping matrix element, and the sum P ͗ij͘ is restricted to nearest neighbors. Despite its simple structure, the solution of this model turns out to be an extremely difficult many-body problem. The situation is particularly complicated near the metalinsulator transition, where U and the bandwidth are roughly of the same order and perturbative schemes (in U or t) are not applicable.With the recent development of the dynamical mean field theory (DMFT) [7-9] a very detailed analysis of the phase diagram of the infinite-dimensional Hubbard model became possible. The iterative perturbation theory (IPT) results of [9] gave a first order metal-insulator transition at finite temperatures. The transition occurs within a coexistence region of metallic and insulating solutions extending from T 0 up to T ‫ء‬ ഠ 0.02W (W : bandwidth). On approaching the metal-insulator transition from the metallic side (i.e., on increasing U), the authors of [9] found a quasiparticle peak with vanishing spectral weight which becomes isolated from the upper and lower Hubbard bands. A consequence of this result is that the opening of the gap and the vanishing of the quasiparticle peak do not happen at the same critical U. The possibility of this scenario was questioned by various authors [2,10-12]. The criticism is partly based on the fact that the IPT is essentially a second order perturbation theory in U (although iterated due to the selfconsistency appearing in the DMFT), whereas the metalinsulator transition happens at U values of the order of the bandwidth.Nonperturbative methods are clearly needed to clarify the situation. At finite temperatures, the quantum...

We present a new method to calculate directly the one-particle self-energy of an impurity Anderson model with Wilson's numerical Renormalization Group method by writing this quantity as the ratio of two correlation functions. This way of calculating Σ(z) turns out to be considerably more reliable and accurate than via the impurity Green's function alone. We show results for the self-energy for the case of a constant coupling between impurity and conduction band (ℑm∆(ω + i0 + ) = const) and the effective ∆(z) arising in the Dynamical Mean Field Theory of the Hubbard model. Implications to the problem of the metal-insulator transition in the Hubbard model are also discussed.

We use the numerical renormalization group method to study an Anderson impurity in a conduction band with the density of states varying as ρ(ω) ∝ |ω| r with r > 0. We find two different fixed points: a local-moment fixed point with the impurity effectively decoupled from the band and a strong-coupling fixed point with a partially screened impurity spin. The specific heat and the spin-susceptibility show powerlaw behaviour with different exponents in strong-coupling and localmoment regime. We also calculate the impurity spectral function which diverges (vanishes) with |ω| −r (|ω| r ) in the strong-coupling (local moment) regime.PACS 75.20.Hr

Wilson's numerical renormalization group (NRG) method for the calculation of dynamic properties of impurity models is generalized to investigate the effective impurity model of the dynamical mean field theory at finite temperatures. We calculate the spectral function and self-energy for the Hubbard model on a Bethe lattice with infinite coordination number directly on the real frequency axis and investigate the phase diagram for the Mott-Hubbard metal-insulator transition. While for T < Tc ≈ 0.02W (W : bandwidth) we find hysteresis with first-order transitions both at Uc1 (defining the insulator to metal transition) and at Uc2 (defining the metal to insulator transition), at T > Tc there is a smooth crossover from metallic-like to insulating-like solutions.71.10. Fd, 71.30.+h

We discuss models of interacting magnetic impurities coupled to a metallic host. If twice the sum of the impurity spins is larger than the total number of host screening channels, the system shows one or more quantum phase transitions where the ground-state spin changes as a function of the inter-impurity couplings. The simplest example is realized by two spin-1/2 Kondo impurities coupled to a single orbital of the host; this model exhibits a singlet-doublet transition. We investigate the phase diagram and crossover behavior of this model and present Numerical Renormalization Group results together with general arguments showing that the quantum phase transition is either of first order or of the Kosterlitz-Thouless type, depending on the symmetry of the Kondo couplings. Connections to other models and possible applications are discussed.Comment: 4 pages, 3 figs. (v2) Minor changes and clarifications, small asymmetry case discussed; application to multilevel quantum dots emphasized, in particular to the experiment of W. G. van der Wiel et al., cond-mat/0110432, where indications for a Kosterlitz-Thouless-type transition in a multilevel dot are found. Final version as publishe

We present a detailed description of the recently proposed numerical renormalization group method for models of quantum impurities coupled to a bosonic bath. Specifically, the method is applied to the spin-boson model, both in the Ohmic and sub-Ohmic cases. We present various results for static as well as dynamic quantities and discuss details of the numerical implementation, e.g., the discretization of a bosonic bath with arbitrary continuous spectral density, the suitable choice of a finite basis in the bosonic Hilbert space, and questions of convergence w.r.t. truncation parameters. The method is shown to provide high-accuracy data over the whole range of model parameters and temperatures, which are in agreement with exact results and other numerical data from the literature.