Impurities on surfaces experience a geometric symmetry breaking induced not only by the on-site crystal-field splitting and the orbital-dependent hybridization, but also by different screening of the Coulomb interaction in different directions. We present a many-body study of the Anderson impurity model representing a Co adatom on graphene, taking into account all anisotropies of the effective Coulomb interaction, which we obtained by the constrained random-phase approximation. The most pronounced differences are naturally displayed by the many-body self-energy projected onto the single-particle states. For the solution of the Anderson impurity model and analytical continuation of the Matsubara data, we employed new implementations of the continuous-time hybridization expansion quantum Monte Carlo and the stochastic optimization method, and we verified the results in parallel with the exact diagonalization method.
Impurities on surfaces experience a geometric symmetry breaking induced not only by the on-site crystal-field splitting and the orbital-dependent hybridization, but also by different screening of the Coulomb interaction in different directions. We present a many-body study of the Anderson impurity model representing a Co adatom on graphene, taking into account all anisotropies of the effective Coulomb interaction, which we obtained by the constrained random-phase approximation. The most pronounced differences are naturally displayed by the many-body self-energy projected onto the single-particle states. For the solution of the Anderson impurity model and analytical continuation of the Matsubara data, we employed new implementations of the continuous-time hybridization expansion quantum Monte Carlo and the stochastic optimization method, and we verified the results in parallel with the exact diagonalization method.
We introduce a block Lanczos (BL) recursive technique to construct quasi-one-dimensional models, suitable for density-matrix renormalization group (DMRG) calculations, from single- as well as multiple-impurity Anderson models in any spatial dimensions. This new scheme, named BL-DMRG method, allows us to calculate not only local but also spatially dependent static and dynamical quantities of the ground state for general Anderson impurity models without losing elaborate geometrical information of the lattice. We show that the BL-DMRG method can be easily extended to treat a multiorbital Anderson impurity model where not only inter- and intraorbital Coulomb interactions but also Hund's coupling and pair hopping interactions are included. We also show that the symmetry adapted BL bases can be utilized, when it is appropriate, to reduce the computational cost. As a demonstration, we apply the BL-DMRG method to three different models for graphene with a structural defect and with a single hydrogen or fluorine absorbed, where a single Anderson impurity is coupled to conduction electrons in the honeycomb lattice. These models include (i) a single adatom on the honeycomb lattice, (ii) a substitutional impurity in the honeycomb lattice, and (iii) an effective model for a single carbon vacancy in graphene. Our analysis of the local dynamical magnetic susceptibility and the local density of states at the impurity site reveals that, for the particle-hole symmetric case at half-filling of electron density, the ground state of model (i) behaves as an isolated magnetic impurity with no Kondo screening, while the ground state of the other two models forms a spin-singlet state where the impurity moment is screened by the conduction electrons. We also calculate the real-space dependence of the spin-spin correlation functions between the impurity site and the conduction sites for these three models. Our results clearly show that, reflecting the presence or absence of unscreened magnetic moment at the impurity site, the spin-spin correlation functions decay as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}3}$, differently from the noninteracting limit ($\ensuremath{\propto}$ ${r}^{\ensuremath{-}2}$), for model (i) and as $\ensuremath{\propto}$ ${r}^{\ensuremath{-}4}$, exactly the same as the noninteracting limit, for models (ii) and (iii) in the asymptotic r, where r is the distance between the impurity site and the conduction site. Finally, based on our results, we shed light on recent experiments on graphene where the formation of local magnetic moments as well as the Kondo-like behavior have been observed.
Recent developments in methods and computational power render it possible to realistically simulate nanoscopic systems such as surfaces, two-dimensional materials, and nanodots including strong electronic correlations. Nanoscopic structuring enables the tailoring of the electronic structure which can be the basis of future electronic devices. This thesis addresses method developments and applications at the interface of ab-initio methods and model based many-body methods for the case of nanoscopically structured systems with strong correlations. In contrast to bulk materials, low-dimensional materials exhibit long-range interactions due to reduced screening. In this work, the general question how these long-range interactions affect electronic properties is investigated. To this end, a variational approach which approximates models with long-range interactions by models with only local interactions is introduced. For the case of an ab-initio derived model of graphene it is found that nonlocal interactions stabilize the semimetallic phase. The quality of this approach is discussed using a simple test case. Realistic models of interacting impurities embedded in an extended solid involve a large amount of bath sites and low symmetries regarding the impurity, which renders an exact treatment impossible. A variational algorithm is presented which optimizes corresponding exactly solvable effective models. Thus, the method is a proposition to an unambiguous solution of the so-called bath-discretization problem in exact diagonalization approaches to the Anderson impurity model. The method is benchmarked for a simple test case and applied to realistic models of Co atoms in Cu hosts and Fe atoms on alkali surfaces. Finally, the (001) surface of Cr is investigated by incorporating local correlation effects into a material realistic description derived from density functional theory. To this end, the LDA DMFT method is used to calculate spectral functions, which are compared to spectroscopic experimental data. So far open experimental features are thereby clarified. Cr(001) exemplifies a situation where correlation effects are determined by the geometric structure of a material: While correlations effects are weak in bulk Cr, they are key for the electronic structure of the Cr(001) surface.
The Hubbard and extended Hubbard models on the honeycomb lattice can be seen as prototype models of a single-layer graphene placed in a high dielectric constant environment that screens the Coulomb interaction. Taking advantage of the absence of a sign problem at half-filling, we study this problem with clusters up to 96 sites with the determinant quantum Monte Carlo method as an impurity solver for the dynamical cluster approximation at finite temperatures. After determining the stability of the semimetallic phase to interaction-induced spin-density wave (SDW), charge-density wave (CDW), and Mott insulating phases, we study the single-particle dynamics of the Dirac fermions. We show that when spontaneous symmetry breaking is avoided, the semimetallic phase is a stable Fermi liquid in the presence of repulsive interactions and that Kondo screening dominates the low temperature regime, even though there is a $\ensuremath{\rho}(\ensuremath{\omega})=|\ensuremath{\omega}|$ type local density of states. We also investigate the impact of the correlation effects on the renormalization of the Fermi velocity vF. We find that vF is not renormalized when only on-site repulsion U is present, but that near-neighbor repulsion V does renormalize vF. This may explain the variations between different measurements of vF in graphene.
The many-body correlation between electrons is the origin of many fascinating phenomena in condensed matter systems, such as high temperature superconductivity, superfluidity, fractional quantum Hall effect, and Mott insulator. Strongly correlated systems have been an important subject of condensed matter physics for several decades, especially after the discovery of high temperature cuprate superconductors. In this thesis, we apply first principles quantum Monte Carlo (QMC) method to several representative systems to study the electron correlations in transition metal oxides (vanadium dioxide) and low dimensional electronic systems (graphene and graphene-like two dimensional systems). Vanadium dioxide (VO2) is a paradigmatic example of a strongly correlated system that undergoes a metal-insulator transition at a structural phase transition. To date, this transition has necessitated significant post-hoc adjustments to theory in order to be described properly. We apply first principles quantum Monte Carlo (QMC) to study the structural dependence of the properties of VO2. Using this technique, we simulate the interactions between electrons explicitly, which allows for the metal-insulator transition to naturally emerge, importantly without ad-hoc adjustments. The QMC calculations show that the structural transition directly causes the metal-insulator transition and a change in the coupling of vanadium spins. This change in the spin coupling results in a prediction of a momentum-independent magnetic excitation in the insulating state. While two-body correlations are important to set the stage for this transition, they do not change significantly when VO2 becomes an insulator. These results show that it is now possible to account for electron correlations in a quantitatively accurate way that is also specific to materials. Electron correlation in graphene is unique because of the interplay of the Dirac cone dispersion of pi electrons with long range Coulomb interaction. The random phase approximation predicts no metallic screening at long distances and low energies because of the zero density of states at Fermi level. It is thus interesting to see how screening takes place in graphene at different length scales. We addressed this problem by computing the structure factor S(q) and S(q, ω) of freestanding graphene using ab initio fixed-node diffusion Monte Carlo and the random phase approximation. The X-ray measured structure factor is reproduced very accurately using both techniques, provided that sigma-bonding electrons are included in the simulations. Strong dielectric screening from sigma electrons are observed, which redshifts the pi plasmons resonance frequency at long distance and reduces the effective interactions between pi electrons at short distance. The short distance screening makes suspended graphene a weakly correlated semimetal which otherwise would be an insulator. The third piece of works is dedicated to studying the low energy excitation of many-body systems using extended Koopmans' theorem (EKT). The EKT provides a straight forward way to compute charge excitation spectra, such as ionization potentials, electron affinities from any level of theory. We implemented the EKT within the QMC framework, and performed systematic benchmark studies of ionization potentials of the second- and third-row atoms, and closed- and open-shell molecules. We also applied it to compute the quasiparticle band structure of solids (graphene). For complex correlated systems, identifying relevant low energy physics degrees of freedom is extremely important to understanding the system's collective behavior at different length scales. In this sense, bridging the realistic systems to lower energy effective lattice models that involve fewer but important degrees of freedom is significant to understanding correlated systems. We have formulated three ab initio density matrix based downfolding (AIDMD) methods to downfold the ab initio systems into effective lattice models. We have demonstrated the successfulness of these methods by applying them to molecules (H2) and periodic systems (hydrogen chain and graphene).
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.
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.