In this paper we review recent developments towards a realistic description of the electronic structure and magnetism of correlated nanosystems. A new class of so-called continuous-time solvers for the quantum impurity problem is discussed, which provides a numerically exact solution without systematic errors due to imaginary time discretization. These solvers are able to handle general interactions, like the full Coulomb vertex. We further show how four-point or higher-order correlation functions of the impurity problem can be computed. This allows the calculation of dynamical susceptibilities which provide information about spin excitations. Moreover, we discuss a principally new many-body scheme recently proposed for the description of non-local correlations in strongly correlated systems. This approach provides a basis for a many-body description of extended correlated nanostructures on a substrate.
In this paper we review recent developments towards a realistic description of the electronic structure and magnetism of correlated nanosystems. A new class of so-called continuous-time solvers for the quantum impurity problem is discussed, which provides a numerically exact solution without systematic errors due to imaginary time discretization. These solvers are able to handle general interactions, like the full Coulomb vertex. We further show how four-point or higher-order correlation functions of the impurity problem can be computed. This allows the calculation of dynamical susceptibilities which provide information about spin excitations. Moreover, we discuss a principally new many-body scheme recently proposed for the description of non-local correlations in strongly correlated systems. This approach provides a basis for a many-body description of extended correlated nanostructures on a substrate.
Nanostructures with open shell transition metal or molecular constituents host often strong electronic correlations and are highly sensitive to atomistic material details. This tutorial review discusses method developments and applications of theoretical approaches for the realistic description of the electronic and magnetic properties of nanostructures with correlated electrons. First, the implementation of a flexible interface between density functional theory and a variant of dynamical mean field theory (DMFT) highly suitable for the simulation of complex correlated structures is explained and illustrated. On the DMFT side, this interface is largely based on recent developments of quantum Monte Carlo and exact diagonalization techniques allowing for efficient descriptions of general four fermion Coulomb interactions, reduced symmetries and spin-orbit coupling, which are explained here. With the examples of the Cr (001) surfaces, magnetic adatoms, and molecular systems it is shown how the interplay of Hubbard U and Hund’s J determines charge and spin fluctuations and how these interactions drive different sorts of correlation effects in nanosystems. Non-local interactions and correlations present a particular challenge for the theory of low dimensional systems. We present our method developments addressing these two challenges, i.e., advancements of the dynamical vertex approximation and a combination of the constrained random phase approximation with continuum medium theories. We demonstrate how non-local interaction and correlation phenomena are controlled not only by dimensionality but also by coupling to the environment which is typically important for determining the physics of nanosystems.
This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems. Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers. Quantum impurity models A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.
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.
In this lecture we give an introduction to theoretical description of strongly correlated materials based on dynamical mean-¿eld theory (DMFT) and its extensions.The goal of this theoretical construction is to retain the many-body aspects of local atomic physics within the extended solid.The effects of short-range non-local correlations within cluster extensions of the DMFT scheme, as well as long-range Àuctuations within the fully renormalized dual-fermion perturbation scheme are discussed extensively.Recent progress in the numerical solution of the DMFT effective quantum impurity problem within the recently developed continuous-time Quantum Monte Carlo schemes is reviewed.We then describe realistic extensions of this approach which combine the accuracy of ¿rst-principle Density-Functional Theory with the treatment of local many-body effects within DMFT.
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.
We discuss the efficient implementation of general impurity solvers for dynamical mean-field theory. We show that both Lanczos and quantum Monte Carlo in different flavors (Hirsch-Fye, continuous-time hybridization- and interaction-expansion) exhibit excellent scaling on massively parallel supercomputers. We apply these algorithms to simulate realistic model Hamiltonians including the full Coulomb vertex, crystal-field splitting, and spin-orbit interaction. We discuss how to remove the sign problem in the presence of non-diagonal crystal-field and hybridization matrices. We show how to extract the physically observable quantities from imaginary time data, in particular correlation functions and susceptibilities. Finally, we present benchmarks and applications for representative correlated systems.