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.

Quantum cluster approaches offer new perspectives to study the complexities of macroscopic correlated fermion systems. These approaches can be understood as generalized mean-field theories. Quantum cluster approaches are non-perturbative and are always in the thermodynamic limit. Their quality can be systematically improved, and they provide complementary information to finite size simulations. They have been studied intensively in recent years and are now well established. After a brief historical review, this article comparatively discusses the nature and advantages of these cluster techniques. Applications to common models of correlated electron systems are reviewed. 1

We introduce an extension of the dynamical mean field approximation (DMFA) which retains the causal properties and generality of the DMFA, but allows for systematic inclusion of non-local corrections. Our technique maps the problem to a self-consistently embedded cluster. The DMFA (exact result) is recovered as the cluster size goes to one (infinity). As a demonstration, we study the Falicov-Kimball model using a variety of cluster sizes. We show that the sum rules are preserved, the spectra are positive definite, and the non-local correlations suppress the CDW transition temperature.Introduction. Strongly interacting electron systems have been on the forefront of theoretical and experimental interest for several decades. This interest has intensified with the discovery of a variety of Heavy Fermion and related non Fermi liquid systems and the high-T c superconductors. In all these systems strong electronic interactions play a dominant role in the selection of at least the low temperature phase. The simplest theoretical models of strongly correlated electrons, the Hubbard model (HM) and the periodic Anderson model (PAM), have remained unsolved in more than one dimension despite a multitude of sophisticated techniques introduced since the inception of the models.With the ground breaking work by Metzner and Vollhardt [1] it was realized that these models become significantly simpler in the limit of infinite dimensions, D = ∞. Namely, provided that the kinetic energy is properly rescaled as 1/ √ D, they retain only local, though nontrivial dynamics: The self energy is constant in momentum space, though it has a complicated frequency dependence. Consequently, the HM and PAM map onto a generalized single impurity Anderson model. The thermodynamics and phase diagram have been obtained numerically by quantum Monte Carlo (QMC) and other methods. [2][3][4] The name dynamical mean field approximation (DMFA) has been coined for approximations in which a purely local self energy (and vertex function) is assumed in the context of a finite dimensional electron system. While it has been shown that this approximation captures many key features of strongly correlated systems even in a finite dimensional context, the DMFA, which leads to an effective single site theory, has some obvious limitations. For example, the DMFA can not describe phases with explicitly nonlocal order parameters, such as d-wave superconductivity, nor can it describe

We present release 1.3 of the ALPS (Algorithms and Libraries for Physics Simulations) project, an international open source software project to develop libraries and application programs for the simulation of strongly correlated quantum lattice models such as quantum magnets, lattice bosons, and strongly correlated fermion systems. Development is centered on common XML and binary data formats, on libraries to simplify and speed up code development, and on full-featured simulation programs. The programs enable non-experts to start carrying out numerical simulations by providing basic implementations of the important algorithms for quantum lattice models: classical and quantum Monte Carlo (QMC) using non-local updates, extended ensemble simulations, exact and full diagonalization (ED), as well as the density matrix renormalization group (DMRG). Changes in the new release include a DMRG program for interacting models, support for translation symmetries in the diagonalization programs, the ability to define custom measurement operators, and support for inhomogeneous systems, such as lattice models with traps. The software is available from our web server at http://alps.comp-phys.org/.

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.