In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method 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 method was 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 for calculating 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.
The Anderson Impurity Model is a generic model to describe systems in which one or more energy levels are coupled to an electronic host. On the energy levels the electrons are subject to a Coulomb interaction. Possible applications of this model are magnetic atoms in a metal or quantum dot systems. Already for one impurity it displays very interesting physics including the well known Kondo effect. For two and more impurities the properties of the system in the local moment regime are determined by an interplay of the Kondo effect and the coupling between the impurity spins. Due to the interplay of multiple energy scales, this model poses a serious task for any computational tool. For a single impurity the model can be very efficiently treated using Wilson"s numerical renormalization group (NRG). Since the numerical effort of NRG calculations rises exponentially with the number of impurities, generically it cannot be applied to systems with more than two impurities. The functional renormalization group (fRG) has been very successfully applied to quantum impurity problems in the past. In this work it will be used to calculate the single-particle dynamics for Anderson-like quantum impurity systems in equilibrium. A comparison to NRG in the case of a single impurity will serve as a benchmark before systems with two impurities and quantum dot systems with two and three dots will be studied. The interplay between the Kondo effect and magnetic and orbital ordering will be investigated in these systems. Although the truncation in the presented fRG scheme is motivated by second order perturbation theory, it will be shown that fRG works also in those cases where perturbation theory fails. Generally it will turn out that fRG is capable of treating a broad variety of Anderson impurity systems within the same framework, including asymmetric impurity levels and interaction between multiple impurities.
In this thesis, we study two recently developed methods to tackle low-dimensional correlated quantum systems.\\In the first part, we benchmark the extension of the functional renormalization group to spin-systems. We apply it to the two-dimensional XXZ model and reproduce the prediction for the phase transition from planar to axial ordering at the isotropic point. The interpretation of the critical scale (where the flow of the susceptibility diverges) as the critical temperature of the system can be questioned, since it yields only good results in the Ising limit. Especially near the isotropic point, this interpretation becomes unsatisfactory as the Mermin-Wagner theorem is violated. We discuss several problems of the method and conclude that it should only be used to explore phase diagrams. In the second part, we extend previous works to two-level quantum dots in the Coulomb blockade regime with special hopping matrices in nonequilibrium, e.g., the Kondo model, to the generic form, including ferromagnetic leads, spin-orbit interactions etc. The dot and the transport observables are determined completely by the hybridization matrix, leading to one of our major results that all these models can be mapped to the Anderson impurity model with ferromagnetic leads. We investigate this model with a well-controlled real-time renormalization group approach and justify the results of a poor man's scaling analysis. By using a singular value decomposition of the tunneling matrix we can rotate the model to the anisotropic Kondo model in the high-energy regime to solve the flow equations analytically. With this, we calculate the stationary dot magnetization and the current. The minimum of the magnetization is found to be an ellipsoid as function of the magnetic field, where the stretching factor determines the distance to the scaling limit. Afterwards, we consider the special case of two external reservoirs and the system being in the scaling limit and discuss the golden-rule as well as the quantum-interference regime in detail. We can give a recipe how to extract the effective model parameters by measuring the current through the dot. Finally, the case of more than two dot-levels is discussed, where our approach becomes inapplicable.
The numerical renormalization group (NRG) approach was developed to identify and quantify different equilibrium regimes of quantum impurity systems (QISs) with unprecedented accuracy by a tailored finite size representation. Out of equilibrium, the steady-state density operator is not of the Boltzmannian form but one that is determined by the imposed boundary conditions. We extend the NRG to the nonequilibrium setting by augmenting each Wilson site with a reservoir, whose coupling functions are calculated via a continuous fraction expansion in order to recover the continuum limit exactly. The nonequilibrium parameters such as a finite bias as well as a finite temperature gradient enters through the Bloch-Redfield tensor (BRT), whose zero eigenvector gives the steady-state density operator. We used the resulting open chain full density matrix (OC-FDM) approach with an effective single lead description to investigate the charge and spin transport through a quantum dot (QD) under finite bias and temperature gradient. The influence of lead asymmetry and an external magnetic field on the transport properties are also studied. For completeness, we have also investigated local properties of the QD, such as charge fluctuations and find excellent agreement with real-time quantum Monte Carlo (RT-QMC) data. The OC-FDM approach was able to explore Kondo energy scales as low as
TK/D≈10−8 in the non-equilibrium regime, as well as show convergence with the established equilibrium benchmarks, such as a quantitative agreement with full density matrix numerical renormalization group (FDM-NRG) and Fermi-liquid scaling at small bias and temperature. Owing to the effective single lead description, a single OC-FDM data point takes orders of magnitude less time on a standard laptop, compared to other state-of-the-art numerical methods.
Numerical renormalization-group (NRG) calculations of quantum impurity models, based on a logarithmic discretization in energy of electronic or bosonic Hamiltonians, provide a powerful tool to describe physics involving widely separated energy scales, as typically encountered in nanostructures and strongly correlated materials. This main advantage of the NRG was however considered a drawback for resolving sharp spectral features at finite energy, such as dissipative atomic peaks. Surprisingly, we find a bunching of many-body levels in NRG spectra near dissipative resonances, and exploit this by combining the widely used Oliveira's
z trick, using an averaging over few discrete NRG spectra, with an optimized frequency-dependent broadening parameter $b(\ensuremath{\omega})$. This strategy offers a tremendous gain in computational power and extracts all the needed information from the raw NRG data without a priori knowledge of the various energy scales at play. As an application we investigate with high precision the crossover from coherent to incoherent dynamics in the spin boson model.
This thesis contributes to the understanding of impurity models. It is divided into two main parts, with a general introduction given in Part I and the research related to it presented in Part II, with the second part being subdivided into two main projects. In the first project, the influence of two many-body effects, the Kondo effect and the Fermi edge singularity, on the absorption and emission spectra of self-assembled quantum dots (QDs) is examined. Whereas the Kondo effect so far was always examined with transport experiments, we show that it has been observed with optical methods for the first time, by comparing experimental data for the absorption line shapes of QDs to calculations with the numerical renormalization group. We continue by examining a QD with strong optical coupling of the energy levels. The resulting interplay of Rabi-oscillations and Kondo effect leads to a new many-body state, a secondary, outer Kondo effect, with Kondo-like correlations between the spin-Kondo and the trion state. The last work regarding optics at QDs addresses the Fermi edge singularity. We show that for QDs this phenomenon can be described numerically on a quantitative level. The second project concerns transport properties of impurity models. First, we present a comprehensive study of the Kondo effect for an InAs-nanowire QD, a system for which the Kondo effect was observed only a few years ago. The second study regarding transport concerns the Kondo effect in bulk metals with magnetic impurities. Although nowadays the Kondo effect is often studied with QDs, it was discovered for iron impurities in noble metals like gold and silver. However, it was unknown for a long time which exact realization of Kondo model describes these systems. We identify the model by comparing numerical calculations for the magnetoresistivity and the dephasing rate for different models to experimental results. The third work about transport concerns the phenomenon that for a fixed type of Kondo model quantities like the magnetoresistivity or the conductivity, respectively, can be scaled onto a universal curve for different parameters, when energies are rescaled with the the Kondo temperature
TK, since it is the only relevant low energy scale of the problem. For finite bandwidth, however, different definitions of
TK (which coincide in the limit of infinite bandwidth) lead to different
TK-values. We show that with a very common definition of
TK, finite bandwidth, which is always present at numerical calculations, can deteriorate the universality of rescaled curves, and we offer an alternative definition of
TK which ensures proper scaling. In the last study presented in this thesis we calculate the Fermi-liquid coefficients for fully screened multi-channel Kondo models. For temperatures below
TK, these models show Fermi-liquid behavior, and the impurity density of states and certain quantities which depend on it, like resistivity, show quadratic dependencies on parameters like temperature or magnetic field, described by the Fermi-liquid coefficients. We calculate these coefficients both analytically and numerically.