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- Renormierungsgruppe (4)
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- Investigation of the microscopic behavior of Mott insulators by means of the density functional theory and many-body methods (2012)
- The objective of this work is twofold. First, we explore the performance of the density functional theory (DFT) when it is applied to solids with strong electronic correlations, such as transition metal compounds. Along this direction, particular effort is put into the refinement and development of parameterization techniques for deriving effective models on a basis of DFT calculations. Second, within the framework of the DFT, we address a number of questions related to the physics of Mott insulators, such as magnetic frustration and electron-phonon coupling (Cs2CuCl4 and Cs2CuBr4), high-temperature superconductivity (BSCCO) and doping of Mott insulators (TiOCl). In the frustrated antiferromagnets Cs2CuCl4 and Cs2CuBr4, we investigate the interplay between strong electronic correlations and magnetism on one hand and electron-lattice coupling on the other as well as the effect of this interplay on the microscopic model parameters. Another object of our investigations is the oxygen-doped cuprate superconductor BSCCO, where nano-scale electronic inhomogeneities have been observed in scanning tunneling spectroscopy experiments. By means of DFT and many-body calculations, we analyze the connection between the structural and electronic inhomogeneities and the superconducting properties of BSCCO. We use the DFT and molecular dynamic simulations to explain the microscopic origin of the persisting under doping Mott insulating state in the layered compound TiOCl.

- Excitations of interacting fermions in reduced dimensions (2010)
- In this thesis, we study the properties of excitations in the systems of interacting fermions. These excitations can be bosonic such as collective modes which we handle in the first part of this thesis or fermionic like quasi particles and quasi holes. One of the important points, to investigate the excitations is their damping which corresponds to their life-time in the system. This thesis consists of two parts, where in both parts, we use the field-theoretical methods to examine the problem.

- A numerical renormalization group approach to dissipative quantum impurity systems (2011)
- The miniaturization of electronics is reaching its limits. Structures necessary to build integrated circuits from semiconductors are shrinking and could reach the size of only a few atoms within the next few years. It will be at the latest at this point in time that the physics of nanostructures gains importance in our every day life. This thesis deals with the physics of quantum impurity models. All models of this class exhibit an identical structure: the simple and small impurity only has few degrees of freedom. It can be built out of a small number of atoms or a single molecule, for example. In the simplest case it can be described by a single spin degree of freedom, in many quantum impurity models, it can be treated exactly. The complexity of the description arises from its coupling to a large number of fermionic or bosonic degrees of freedom (large meaning that we have to deal with particle numbers of the order of 10^{23}). An exact treatment thus remains impossible. At the same time, physical effects which arise in quantum impurity systems often cannot be described within a perturbative theory, since multiple energy scales may play an important role. One example for such an effect is the Kondo effect, where the free magnetic moment of the impurity is screened by a "cloud" of fermionic particles of the quantum bath. The Kondo effect is only one example for the rich physics stemming from correlation effects in many body systems. Quantum impurity models, and the oftentimes related Kondo effect, have regained the attention of experimental and theoretical physicists since the advent of quantum dots, which are sometimes also referred to as as artificial atoms. Quantum dots offer a unprecedented control and tunability of many system parameters. Hence, they constitute a nice "playground" for fundamental research, while being promising candidates for building blocks of future technological devices as well. Recently Loss' and DiVincenzo's p roposal of a quantum computing scheme based on spins in quantum dots, increased the efforts of experimentalists to coherently manipulate and read out the spins of quantum dots one by one. In this context two topics are of paramount importance for future quantum information processing: since decoherence times have to be large enough to allow for good error correction schemes, understanding the loss of phase coherence in quantum impurity systems is a prerequisite for quantum computation in these systems. Nonequilibrium phenomena in quantum impurity systems also have to be understood, before one may gain control of manipulating quantum bits. As a first step towards more complicated nonequilibrium situations, the reaction of a system to a quantum quench, i.e. a sudden change of external fields or other parameters of the system can be investigated. We give an introduction to a powerful numerical method used in this field of research, the numerical renormalization group method, and apply this method and its recent enhancements to various quantum impurity systems. The main part of this thesis may be structured in the following way: - Ferromagnetic Kondo Model, - Spin-Dynamics in the Anisotropic Kondo and the Spin-Boson Model, - Two Ising-coupled Spins in a Bosonic Bath, - Decoherence in an Aharanov-Bohm Interferometer.

- Spin-wave calculations for Heisenberg magnets with reduced symmetry (2011)
- The phenomenon of magnetism is a pure quantum effect and has been studied since the beginning of civilization. The practical use of magnetic materials for technical purposes was well established in the 19th century; still nowadays there is no lack of new high-tech applications based on magnetism for example in information technology to store and process data. This thesis does not focus on the development of new applications of magnetism in technology, nor enhancement of known fields of application. Instead, the intention is to use a quantum theory of magnetism for obtaining new insights on physical effects that accompany the phenomenon of magnetism. Therefore three different model systems, each of which are believed to describe a class of real compounds, are considered. Starting from the idea that magnetism can be understood by use of the so-called Heisenberg model that microscopically characterizes the interaction between localized magnetic moments, we restrict ourselves to the case where a long-range magnetic order is present. In order to deduce consequences resulting from this microscopic picture we use the spin-wave theory that is introduced in the first chapter. Central objects of this theory are the magnons which are elementary quantum excitations in ordered magnets. An application of these mathematical techniques to a model that describes an antiferromagnet in an external magnetic field is presented in the second chapter. Quantities like the spin-wave velocity and the damping of magnons are calculated using a Hermitian operator approach in the framework of spin-wave theory. A strong renormalization of the magnetic excitations arises because the symmetry of the system is reduced due to the external magnetic field. In the second model system, that describes thin films of a ferromagnet, concepts of classical physics meet quantum physics: The magnetic dipole-dipole interaction that is also known in everyday life from the magnetic forces between magnets and was initially formulated in the theory of electromagnetism, is included in the microscopic model. Having a special compound in mind where the magnetic excitations are directly accessible in experiments, the energy dispersions of magnon modes in thin-film ferromagnets are deduced. Our approach is essentially a basis for further investigations beyond this thesis to describe strong correlations and condensation of magnons. A recent realization of data processing devices with spin waves puts the understanding of physical processes in these ferromagnetic films in the focus of upcoming research. The third model system brings in the so-called frustration where the interactions between the spins are such that the total energy cannot be minimized by an appropriate alignment of the magnetic moments in the classical picture. In the simplest case this appears because the antiferromagnetically coupled spins are located on a triangular lattice. This situation will lead to strong quantum fluctuations which make this model system interesting. Finally the overall symmetry is reduced by inclusion of spin anisotropies and an external magnetic field. Instead of focusing on the properties of the magnetic excitations, the effect of the magnetic field on the properties of the lattice vibrations is subject to the investigation. This is interesting because the characteristics of lattice vibrations can be measured experimentally using the supersonic technique.

- Applications of the functional renormalization group to quantum liquids (2007)
- The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.

- Spin-wave calculations for low-dimensional magnets (2006)
- The focus of this thesis is on quantum Heisenberg magnets in low dimensions. We modify the method of spin-wave theory in order to address two distinct issues. In the first part we develop a variant of spin-wave theory for low-dimensional systems, where thermodynamic observables are calculated from the Gibbs free energy for fixed order parameter. We are able to go beyond linear spin-wave theory and systematically calculate two-loop correction to the free energy. We use our method to determine the low-temperature physics of Heisenberg ferromagnets in one, two and three spatial dimensions. In the second part of the thesis, we treat a two-dimensional Heisenberg antiferromagnet in the presence of a uniform external magnetic field. We determine the low-temperature behavior of the magnetization curve within spin-wave theory by taking the absence of the spontaneous staggered magnetization into account. Additionally, we perform quantum Monte Carlo simulations and subsequently show that numerical findings are qualitatively comparable to spin-wave results. Finally, we apply our method to an experimentally motivated case of the distorted honeycomb lattice in order to determine the strength of the exchange interactions.

- Aspects of strong correlations in low dimensions (2005)
- The challenging intricacies of strongly correlated electronic systems necessitate the use of a variety of complementary theoretical approaches. In this thesis, we analyze two distinct aspects of strong correlations and develop further or adapt suitable techniques. First, we discuss magnetization transport in insulating one-dimensional spin rings described by a Heisenberg model in an inhomogeneous magnetic field. Due to quantum mechanical interference of magnon wave functions, persistent magnetization currents are shown to exist in such a geometry in analogy to persistent charge currents in mesoscopic normal metal rings. The second, longer part is dedicated to a new aspect of the functional renormalization group technique for fermions. By decoupling the interaction via a Hubbard-Stratonovich transformation, we introduce collective bosonic variables from the beginning and analyze the hierarchy of flow equations for the coupled field theory. The possibility of a cutoff in the momentum transfer of the interaction leads to a new flow scheme, which we will refer to as the interaction cutoff scheme. Within this approach, Ward identities for forward scattering problems are conserved at every instant of the flow leading to an exact solution of a whole hierarchy of flow equations. This way the known exact result for the single-particle Green's function of the Tomonaga-Luttinger model is recovered.

- Renormalization-group approach to the spectral function of the Tomonaga-Luttinger model (2003)
- Die Entwicklung der Renormierungsgruppen-Technik, die in ihrer feldtheoretischen Version auf Ideen von Stückelberg und Petermann und in der Festkörperphysik auf K.G. Wilson zurückgeht, hat wesentliche Einsichten in die Natur physikalischer Systeme geliefert. Insbesondere das Konzept der so genannten Universalitätsklassen erhellt, warum Systeme, die durch scheinbar sehr verschiedene Hamilton-Operatoren beschrieben werden, doch im Wesentlichen die selbe (Niederenergie-)Physik zeigen. Ein weiterer Grund für den Erfolg dieser Methode liegt darin begründet, dass sie in systematischer Weise unendlich viele Feynman-Diagramme aufsummiert und somit über konventionelle Störungstheorie hinaus geht. Dies spielt in der Festkörperphysik vor allem dann eine wichtige Rolle, wenn das vorliegende physikalische System stark korreliert ist. Entsprechend der Vielzahl von Anwendungsmöglichkeiten hat sich in den vergangenen Jahrzehnten eine große Bandbreite verschiedener Formulierungen der Renormierungsgruppen-Technik ergeben. Eine davon ist die sogenannte funktionale Renormierungsgruppe, die auf Wegner und Houghton zurück geht und die auch in der vorliegenden Arbeit benutzt und weiter entwickelt wurde. Wir haben hier insbesondere auf die Einbeziehung der wichtigen Reskalierungsschritte wertgelegt. Als erstes Anwendungsgebiet des neu entwickelten Formalismus wurden stark korrelierte Elektronen in einer Raumdimension ausgewählt und hier insbesondere ein Modell, das als Tomonaga-Luttinger-Modell (TLM) bezeichnet wird. Im TLM wechselwirken Elektronen mit einer strikt linearen Energiedispersion ausschließlich über so genannte Vorwärtsstreu-Prozesse. Aufgrund der Linearisierung der Energiedispersion nahe der Fermipunkte ergibt sich ein Modell, das z.B. mit Hilfe der so genannten Bosonisierungs-Technik exakt gelöst werden kann. Hauptziel der vorliegenden Arbeit ist es, die bekannte Spektralfunktion dieses Modells unter Verwendung des Renormierungsgruppen-Formalismus zu reproduzieren. Gegenüber der bisherigen Implementierung der Renormierungsgruppe, bei der lediglich der Fluss einer endlichen Anzahl von Kopplungskonstanten betrachtet wird, stellt die Berechnung des Flusses ganzer Korrelationsfunktionen eine enorme Erweiterung dar. Der Erfolg dieser Herangehensweise im TLM bestärkt die Hoffnung, dass es in Zukunft auch möglich sein wird, die Spektralfunktionen anderer Modelle mit dieser Methode zu berechnen, bei denen herkömmliche Techniken versagen.

- Application of the Functional Renormalization Group to Bose systems with broken symmetry (2009)
- The physics of interacting bosons in the phase with broken symmetry is determined by the presence of the condensate and is very different from the physics in the symmetric phase. The Functional Renormalization Group (FRG) represents a powerful investigation method which allows the description of symmetry breaking with high efficiency. In the present thesis we apply FRG for studying the physics of two different models in the broken symmetry phase. In the first part of this thesis we consider the classical O(1)-model close to the critical point of the second order phase transition. Employing a truncation scheme based on the relevance of coupling parameters we study the behavior of the RG-flow which is shown to be influenced by competition between two characteristic lengths of the system. We also calculate the momentum dependent self-energy and study its dependence on both length scales. In the second part we apply the FRG-formalism to systems of interacting bosons in the phase with spontaneously broken U(1)-symmetry in arbitrary spatial dimensions at zero temperature. We use a truncation scheme based on a new non-local potential approximation which satisfy both exact relations postulated by Hugenholtz and Pines, and Nepomnyashchy and Nepomnyashchy. We study the RG-flow of the model, discuss different scaling regimes, calculate the single-particle spectral density function of interacting bosons and extract both damping of quasi-particles and spectrum of elementary excitations from the latter.

- A kinetic theory for spin waves in yttrium-iron garnet (2013)
- Spin waves in yttrium-iron garnet has been the subject of research for decades. Recently the report of Bose-Einstein condensation at room temperature has brought these experiments back into focus. Due to the small mass of quasiparticles compared to atoms for example, the condensation temperature can be much higher. With spin-wave quasiparticles, so-called magnons, even room temperature can be reached by externally injecting magnons. But also possible applications in information technologies are of interest. Using excitations as carriers for information instead of charges delivers a much more efficient way of processing data. Basic logical operations have already been realized. Finally the wavelength of spin waves which can be decreased to nanoscale, gives the opportunity to further miniaturize devices for receiving signals for example in smartphones. For all of these purposes the magnon system is driven far out of equilibrium. In order to get a better fundamental understanding, we concentrate in the main part of this thesis on the nonequilibrium aspect of magnon experiments and investigate their thermalization process. In this context we develop formalisms which are of general interest and which can be adopted to many different kinds of systems. A milestone in describing gases out of equilibrium was the Boltzmann equation discovered by Ludwig Boltzmann in 1872. In this thesis extensions to the Boltzmann equation with improved approximations are derived. For the application to yttrium-iron garnet we describe the thermalization process after magnons were excited by an external microwave field. First we consider the Bose-Einstein condensation phenomena. A special property of thin films of yttrium-iron garnet is that the dispersion of magnons has its minimum at finite wave vectors which leads to an interesting behavior of the condensate. We investigate the spatial structure of the condensate using the Gross-Pitaevskii equation and find that the magnons can not condensate only at the energy minimum but that also higher Fourier modes have to be occupied macroscopically. In principle this can lead to a localization on a lattice in real space. Next we use functional renormalization group methods to go beyond the perturbation theory expressions in the Boltzmann equation. It is a difficult task to find a suitable cutoff scheme which fits to the constraints of nonequilibrium, namely causality and the fluctuation-dissipation theorem when approaching equilibrium. Therefore the cutoff scheme we developed for bosons in the context of our considerations is of general interest for the functional renormalization group. In certain approximations we obtain a system of differential equations which have a similar transition rate structure to the Boltzmann equation. We consider a model of two kinds of free bosons of which one type of boson acts as a thermal bath to the other one. Taking a suitable initial state we can use our formalism to describe the dynamics of magnons such that an enhanced occupation of the ground state is achieved. Numerical results are in good agreement with experimental data. Finally we extend our model to consider also the pumping process and the decrease of the magnon particle number till thermal equilibrium is reached again. Additional terms which explicitly break the U(1)-symmetry make it necessary to also extend the theory from which a kinetic equation can be deduced. These extensions are complicated and we therefore restrict ourselves to perturbation theory only. Because of the weak interactions in yttrium-iron garnet this provides already good results.