The single impurity Anderson model (SIAM) is one of the most recurrent model Hamiltonians in condensed matter physics, playing an important role in the description of systems where electron-electron interactions can not be neglected, such as strongly interacting quantum dots and doped metals with magnetic impurities. In order to address the variety of scenarios in which the model can be applied many impurity solvers were developed for the SIAM. It is consensus that there is no universal method that provides reliable results for the model parameter space of experimental interest. The main methods
found in the literature di ers signi cantly on the computational cost involved in the numerical implementation and in the range of the model parameters where the used method is reliable. In this undergraduate dissertation we study the reliability of some impurity solvers for the SIAM, namely, the equations of motion (EOM) method, the numerical renormalization group (NRG), the non-crossing approximation (NCA), and the one crossing approximation (OCA). We study the Green's function of a quantum dot using the aforementioned impurity solvers and then calculate the density of states
as a function of the temperature and the chemical potential of the quantum dot . Our analysis is focused on the Coulomb blockade and the Kondo physics. We report that some standard approximations based on the EOM technique display a rather unexpected poor behaviour in the Coulomb blockade regime even at high temperatures.
The single impurity Anderson model (SIAM) is one of the most recurrent model Hamiltonians in condensed matter physics, playing an important role in the description of systems where electron-electron interactions can not be neglected, such as strongly interacting quantum dots and doped metals with magnetic impurities. In order to address the variety of scenarios in which the model can be applied many impurity solvers were developed for the SIAM. It is consensus that there is no universal method that provides reliable results for the model parameter space of experimental interest. The main methods
found in the literature di ers signi cantly on the computational cost involved in the numerical implementation and in the range of the model parameters where the used method is reliable. In this undergraduate dissertation we study the reliability of some impurity solvers for the SIAM, namely, the equations of motion (EOM) method, the numerical renormalization group (NRG), the non-crossing approximation (NCA), and the one crossing approximation (OCA). We study the Green's function of a quantum dot using the aforementioned impurity solvers and then calculate the density of states
as a function of the temperature and the chemical potential of the quantum dot . Our analysis is focused on the Coulomb blockade and the Kondo physics. We report that some standard approximations based on the EOM technique display a rather unexpected poor behaviour in the Coulomb blockade regime even at high temperatures.
We study the single impurity Anderson model (SIAM) using the equations of motion method (EOM), the non-crossing approximation (NCA), the one-crossing approximation (OCA), and Wilson's numerical renormalization group (NRG). We calculate the density of states and the linear conductance focusing on their dependence on the chemical potential and on the temperature paying special attention to the Kondo and Coulomb blockade regimes for a large range of model parameters. We report that some standard approximations based on the EOM technique display a rather unexpected poor behavior in the Coulomb blockade regime even at high temperatures. Our study offers a critical comparison between the different methods as well as a detailed compilation of the shortcomings and limitations due the approximations involved in each technique, thus allowing for a cost-benefit analysis of the different solvers that considers both numerical precision and computational performance.
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.
O modelo de impureza única de Anderson (SIAM) é um dos modelos mais recorrentes em física da matéria condensada, desempenhando um papel importante na descrição de sistemas onde as interações elétron-elétron não podem ser desprezadas, tais como pontos quânticos fortemente interagentes e metais dopados com impurezas magnéticas. A solução numérica deste modelo também constitui uma etapa essencial para obter as quantidades espectrais de modelos de rede de muitos corpos como o modelo de Hubbard e para obter uma descrição teórica consistente de uma enorme variedade de materiais fortemente correlacionados, no contexto da Teoria de Campo Médio Dinâmica (DMFT) e sua combinação com métodos de cálculo de estrutura eletrônica como a teoria do funcional da densidade (DFT). A aplicabilidade do modelo de Anderson se extende também ao campo muito ativo da eletrônica molecular, onde uma combinação de DFT, DMFT e o formalismo das funções de Green fora do equilíbrio (NEGF) pode ser usada para fornecer uma descrição realista das propriedades de transporte de dispositivos de eletrônica molecular levando em conta os efeitos de correlações fortes quando elas estão presentes. Para abordar tal variedade de cenários muitos métodos numéricos de solução foram desenvolvidos para o modelo de Anderson. É consenso que não existe nenhum método universal que forneça resultados con fiáveis para o espaço de parâmetros do modelo de interesse experimental. Os principais métodos encontrados na literatura diferem signi cativamente quanto ao custo computacional envolvido na implementação numérica e na faixa de parâmetros do modelo onde o método usado é con fiável. Nesta tese nós avaliamos a confi abilidade de alguns métodos numéricos de solução para o SIAM, a saber, o método das equações de movimento (EOM), o grupo de renormalização numérico (NRG), a aproximação de não cruzamento (NCA) e a aproximação de um cruzamento (OCA). Primeiro, nós estudamos função de Green de um ponto quântico usando os métodos supracitados e então calculamos a densidade de estados bem como a condutância à voltagem nula em função da temperatura do ponto quântico no limite de banda larga para uma variedade de cenários. Nossa análise é focada no regime do bloqueio de Coulomb e na física de Kondo. Nós relatamos que algumas aproximações padrão baseadas na técnica EOM exibem um comportamento pobre bastante inesperado no regime de bloqueio de Coulomb até mesmo em temperaturas altas. Em seguida, nós estudamos o modelo de Hubbard aplicado a uma rede hipercúbica e calculamos a densidade de estados do modelo usando a DMFT combinada com EOM, NCA e OCA como métodos para o problema de uma impureza prestando atenção particular à física da transição metal-isolante de Mott. Por meio de um gráfi co de contornos nós também apresentamos uma evolução da densidade de estados a medida que a interação coulombiana é aumentada comparando os principais atributos da transição exibida em cada método. Nós discutimos a aplicabilidade de cada método baseada em nossos resultados e em resultados prévios relatados na literatura ao longo dos anos.
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 the present work, we calculate the conductance through a single quantum dot weakly coupled to metallic contacts. We use the spin-1/2 Anderson model to describe the quantum dot, while considering a finite Coulomb repulsion. We solve the interacting system using the noncrossing approximation (NCA) and the one-crossing approximation (OCA). We obtain the linear response conductance as a function of temperature and energy position of the localized level. From the comparison of both approximations we extract the role of the vertex corrections, which are introduced in the OCA calculations and neglected in the NCA scheme. As a function of the energy position, we observe that the diagrams omitted within the NCA are really important for appropriately describing transport phenomena in Kondo systems as well as in the mixed valence regime. On the other hand, as a function of temperature, the corrections introduced by the OCA partly recover the universal scaling properties known from numerical approaches such as the numerical renormalization group.
In this thesis we deal with the description of quantum impurity systems. In order to achieve such an aim, we exploit in the first part of this work the Reduced Density Matrix (RDM) formalism, in the second one the Density Matrix-Numerical Renormalization Group (DM-NRG) method. The first half of the thesis deals with the derivation and the application of out-of-equilibrium and non-perturbative approximations for the dynamics of the Single Impurity Anderson Model (SIAM) in the perturbative, intermediate coupling and Kondo cross-over regimes. In this part of the thesis, the main aim is to develop new technical tools useful to extend the RDM formalism from the well established perturbative regime to the intermediate and, in the future, to the strong coupling regimes. In this part, we analyse the advantages and drawbacks of the so-called Dressed Second Order (DSO), Dressed DSO (DDSO) and the Resonant Tunnelling Approximation (RTA). In the DSO we include processes capable to describe the broadening of the Coulomb blockade peaks but the DSO itself, as the RTA, is not capable to describe the Kondo resonance width properly. In the DDSO we propose a prescription to solve this issue and to describe properly the exponential dependence of the Kondo resonance width. However, in the DDSO the vertex corrections are not included and this yields a spurious conductance peak in the empty orbital regime (Coulomb valley with zero electrons). For this reason we turn our attention to the inclusion of the vertex corrections within the RTA framework. In the second part, we describe transport and thermodynamic properties of carbon-nanotubes (CNTs) based quantum dots using the DM-NRG method. We present results on the spectral function and the linear conductance in cooperation with Catalin Pascu Moca and Gergely Zarand from the Universities of Oradea and Budapest, respectively. In order to model the CNTs we consider an extended Anderson Hamiltonian. Exploiting such an Hamiltonian we study the crossover from the SU(4) to the SU(2) Kondo effect. We demonstrate numerically and analytically that the Kondo temperature scales inversely with respect to the SU(4)-symmetry breaking parameter Δ, tending to the Kondo temperature of the SU(2) Kondo effect in the large Δ limit. Furthermore we study the CNT quantum dot in the presence of a parallel and perpendicular (with respect to the CNT axis) magnetic field. We find necessary conditions to observe the so-called ``Kondo revivals'' for specific values of the magnetic field. In the final part of this thesis, we model a real world device realised by Jean-Pierre Cleuziou, Ngoc-Viet Nguyen and Wolfgang Wernsdorfer in Grenoble by means of the previously studied Hamiltonian. For this system we search for the best set of parameters to reproduce the experimentally measured linear conductance and we study the underlying Kondo state evaluating the CNT added entropy and specific heat.