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.
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.
An essential role for the description of correlated quantum many-particle systems is played by impurity problems. These consist of a small number of localized orbitals, where the electrons are subject to a Coulomb-interaction, coupled to noninteracting bath degrees of freedom. Impurity models are used for the description of magnetic impurities in metals, of quantum points in nanostructures, and in the context of the dynamical mean field theory (DMFT). For their solution a large number of numerical methods exists. We mention here quantum Monte Carlo methods, the numerical renormalization group, or the exact diagonalization method. In this thesis we introduce a new approach to impurity problems, which is based on the functional renormalization group (fRG). In contrast to the methods mentioned above this approach is approximative, but it can be applied with comparatively lower numerical effort. The aim of this thesis is to examine the advantages and limits of this method. An important role in this connection is played by the single impurity Anderson model (SIAM), which is already understood very well with other methods and is seen as a prototype for models with strong local correlations. The spin fluctuations of this model are governed by the so-called Kondo energy scale and a central question of this thesis is whether this scale can be reproduced with our approach. We begin with a short introduction to the Green's function formalism for correlated quantum many particle systems. Then the fRG flow equations for the one-particle irreducible vertex functions are deduced, and several approximation schemes are discussed. In chapter 4 we introduce the fRG scheme used in this thesis. At first the SIAM is mapped to a semi-infinite chain, in which the interacting orbital is given by the first site of this chain. The system is then subdivided into two parts. The first part, which is called 'core', consists of the interacting orbital and the first L bath sites. The remaining bath sites form the second part. At the beginning both parts are decoupled from each other such that the core can be solved exactly. Starting with this exact solution the coupling between the core and the remaining bath sites is switched on slowly in a renormalization group flow. This way it is possible to calculate the one-particle and two-particle correlation functions on the correlated site. We call this flow scheme 'hybridization flow'. The flow equations are formulated in an effective theory on the first bath site outside the core, which turns out to be advantageous compared to other implementations of the hybridization flow. In chapter 5 this hybridization flow method is applied to the SIAM with semi-elliptic bath density of states. We discuss the differences which arise for different core-sizes (L = 0,1,2,3) and for different truncations of the fRG flow equations. In the cases L=0 and L=2 the local Fermi liquid properties of the SIAM and the associated Kondo scale can not be reproduced in both approximation schemes. For the core sizes L=1 and L=3 the ground state of the core is already a spin singlet state in the beginning of the flow and the Fermi liquid properties are reproduced. The Kondo energy scale, which determines the width of the central quasi-particle peak and the size of the local spin-susceptibility, is however not accurately resolved. This is partly due to the finite temperature used in our implementation such that for larger interaction strength the temperature is above the Kondo energy scale. The application of the hybridization flow method to quantum impurity problems in the context of the DMFT is discussed in chapter 6. In this approach lattice models with a local interaction, like the Hubbard model, are mapped to an effective impurity problem for a single lattice site embedded in a dynamical mean field representing the influence of the other electrons. This scheme is exact in infinitely many dimensions and a nonperturbative approximation method for finite-dimensional systems. In particular the Mott metal-insulator transition in the Hubbard model is successfully described by this method. One class of quantities, which is easily accessible from the hybridization flow scheme, is given by the local two-particle correlation functions. These play an essential role in non-local extensions of the DMFT. Therefore we mainly focus on these quantities. At first we apply our flow scheme to the Hubbard model on the Bethe lattice in infinite dimensions and calculate the local one-particle irreducible vertex functions in the insulating and the metallic phase in good agreement with previous calculations, which use the exact diagonalization method as impurity solver. After this we extend our flow scheme to a cluster DMFT method, which includes short-range antiferromagnetic correlations and calculate again the local and non-local vertex functions for the two-dimensional Hubbard model. We compare these to the results of the single-site DMFT.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
The unusual low temperature behaviour of metals and metal alloys, such as heavy fermion systems, Mott insulators and unconventional superconductors, has been a central topic of scientific investigation for over half a century. Due to strong correlations between the many particles, the solution to most theoretical models of these systems is difficult because standard perturbation methods break down. In certain cases, non-perturbative renormalization group approaches can be used to access the low energy behaviour and take into account the effects of strong correlations. However, these methods do not work for several important and interesting classes of model. These include lattice models with finite dimension, models with impurities of high degeneracy and steady state transport through quantum dots. The aim of this thesis is therefore to develop a new and general renormalization group approach that will lend itself to wide application in models where existing techniques fail. The method explored here utilises renormalized perturbation theory (RPT) in conjunction with scaling equations and collective excitations. It is tested on the single impurity Anderson model, the paradigmatic model used to understand magnetic impurities within a non-magnetic host metal. The Anderson model is simple yet exhibits interesting features due to strong correlation, such as a single renormalized low energy scale called the Kondo temperature. This model is also well understood as it has been solved using the numerical renormalization group (NRG), which provides ample numerical data against which this new technique is compared. Finally, a preliminary study is given for this method when it is applied to a wider class of models.
We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.
We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.
Using the functional renormalization group (FRG) and the numerical renormalization group (NRG), we calculate the spectral function of the Anderson impurity model at zero and finite temperatures. In our FRG scheme spin fluctuations are treated nonperturbatively via a suitable Hubbard-Stratonovich field, but vertex corrections are neglected. A comparison with our highly accurate NRG results shows that this FRG scheme provides a good description of the spectral line shape at zero and finite temperatures both in the weak- and strong-coupling regimes, although at zero temperature the FRG is not able to reproduce the known exponential narrowing of the Kondo resonance at strong coupling.