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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.
Landau's Fermi liquid theory has been the main tool for investigating interactions between fermions at low energies for more than 50 years. It has been successful in describing, amongst other things, the mass enhancement in ³He and the thermodynamics of a large class of metals. Whilst this in itself is remarkable given the phenomenological nature of the original theory, experiments have found several materials, such as some superconducting and heavy-fermion materials, which cannot be described within the Fermi liquid picture. Because of this, many attempts have been made to understand these ''non Fermi liquid'' phases from a theoretical perspective. This will be the broad topic of the first part of this thesis and will be investigated in Chapter 2, where we consider a two-dimensional system of electrons interacting close to a Fermi surface through a damped gapless bosonic field. Such systems are known to give rise to non Fermi liquid behaviour. In particular we will consider the Ising-nematic quantum critical point of a two-dimensional metal. At this quantum critical point the Fermi liquid theory breaks down and the fermionic self-energy acquires the non Fermi liquid like {omega}²/³ frequency dependence at lowest order and within the canonical Hertz-Millis approach to quantum criticality of interacting fermions. Previous studies have however shown that, due to the gapless nature of the electronic single-particle excitations, the exponent of 2/3 is modified by an anomalous dimension {eta_psi} which changes, not only the exponent of the frequency dependence, but also the exponent of the momentum dependence of the self-energy. These studies also show that the usual 1/N-expansion breaks down for this problem. We therefore develop an alternative approach to calculate the anomalous dimensions based on the functional renormalization group, which will be introduced in the introductory Chapter 1. Doing so we will be able to calculate both the anomalous dimension renormalizing the exponent of the frequency dependence and the exponent renormalizing the momentum dependence of the self-energy. Moreover we will see that an effective interaction between the bosonic fields, mediated by the fermions, is crucial in order to obtain these renormalizations.
In the second part of this thesis, presented in Chapter 3, we return to Fermi liquid theory itself. Indeed, despite its conceptual simplicity of expressing interacting electrons through long-lived quasi-particles which behave in a similar fashion as free particles, albeit with renormalized parameters, it remains an active area of research. In particular, in order to take into account the full effects of interactions between quasi-particles, it is crucial to consider specific microscopic models. One such effect, which is not captured by the phenomenological theory itself, is the appearance of non-analytic terms in the expansions of various thermodynamic quantities such as heat-capacity and susceptibility with respect to an external magnetic field, temperature, or momentum. Such non-analyticities may have a large impact on the phase diagram of, for example, itinerant electrons near a ferromagnetic quantum phase transition. Inspired by this we consider a system of interacting electrons in a weak external magnetic field within Fermi liquid theory. For this system we calculate various quasi-particle properties such as the quasi-particle residue, momentum-renormalization factor, and a renormalization factor which relates to the self-energy on the Fermi surface. From these renormalization factors we then extract physical quantities such as the renormalized mass and renormalized electron Lande g-factor. By calculating the renormalization factors within second order perturbation theory numerically and analytically, using a phase-space decomposition, we show that all renormalization factors acquire a non-analytic term proportional to the absolute value of the magnetic field. We moreover explicitly calculate the prefactors of these terms and find that they are all universal and determined by low-energy scattering processes which we classify. We also consider the non-analytic contributions to the same renormalization factors at finite temperatures and for finite external frequencies and discuss possible experimental ways of measuring the prefactors. Specifically we find that the tunnelling density of states and the conductivity acquire a non-analytic dependence on magnetic field (and temperature) coming from the momentum-renormalization factor. For the latter we discuss how this relates to previous works which show the existence of non-analyticities in the conductivity at first order in the interaction.
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.
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a
formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account.
In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0.
Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic
second-magnon regime.
In this work I investigate two different systems - spin systems and charge-density-waves. The same theoretical method is used to investigate both types of system. My investigations are motivated by experimental investigations and the goal is to describe the experimental results theoretically. For this purpose I formulate kinetic equations starting from the microscopical dynamics of the systems.
First of all, a method is formulated to derive the kinetic equations diagrammatically. Within this method an expansion in equal-time connected correlation functions is carried out. The generating functional of connected correlations is employed to derive the method.
The first system to be investigated is a thin stripe of the magnetic insulator yttrium-iron-garnet (YIG). Magnons are pumped parametrically with an external microwave field. The motivation of my theoretical investigations is to explain the experimental observations. In a small parameter range close to the confluence field strength where confluence processes of two parametrically pumped magnons with the same wave vector becomes kinematically possible the efficiency of the pumping is reduced or enhanced depending on the pumping field strength. Because it is expected that that confluence and splitting processes of magnons are essential for the experimental observations I go beyond the kinetic theories that are conventionally applied in the context of parametric excitations in YIG and investigate the influence of cubic vertices on the parametric instability of magnons in YIG.
Furthermore, the influence of phonons is investigated. Usually in the literature these are taken into account as heat bath. Here, I want to explain experiments where an accumulation of magnetoelastic bosons - magnon-phonon-quasi-particles - has been observed. I employ the method of kinetic equations to investigate this phenomenon theoretically. The kinetic theory is able to reproduce the experimental observations and it is shown that the accumulation of magnetoelastic bosons is purely incoherent.
Finally, charge-density waves (CDW) in quasi-one-dimensional materials will be investigated. Charge-density waves emerge from a Peierls-instability and are a prime example for spontaneous symmetry breaking in solids. Again, the motivation for my theoretical investigations are an experiment where the spectrum of amplitude and phase phonon modes has been measured. Starting from the Fröhlich-Hamiltonian I derive kinetic equations and from these kinetic equations the equations of motion for the CDW order parameter can be derived. The frequencies and damping rates of amplitude and phase phonon modes will be derived from the linearized equations of motion. I compare my theory with existing methods. Furthermore, I also investigate the influence of Coulomb interaction.
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.
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.
The present thesis is primarily concerned with the application of the functional renormalization group (FRG) to spin systems. In the first part, we study the critical regime close to the Berezinskii-Kosterlitz-Thouless (BKT) transition in several systems. Our starting point is the dual-vortex representation of the two-dimensional XY model, which is obtained by applying a dual transformation to the Villain model. In order to deal with the integer-valued field corresponding to the dual vortices, we apply the lattice FRG formalism developed by Machado and Dupuis [Phys. Rev. E 82, 041128 (2010)]. Using a Litim regulator in momentum space with the initial condition of isolated lattice sites, we then recover the Kosterlitz-Thouless renormalization group equations for the rescaled vortex fugacity and the dimensionless temperature. In addition to our previously published approach based on the vertex expansion [Phys. Rev. E 96, 042107 (2017)], we also present an alternative derivation within the derivative expansion. We then generalize our approach to the O(2) model and to the strongly anisotropic XXZ model, which enables us to show that weak amplitude fluctuations as well as weak out-of-plane fluctuations do not change the universal properties of the BKT transition.
In the second part of this thesis, we develop a new FRG approach to quantum spin systems. In contrast to previous works, our spin functional renormalization group (SFRG) does not rely on a mapping to bosonic or fermionic fields, but instead deals directly with the spin operators. Most importantly, we show that the generating functional of the irreducible vertices obeys an exact renormalization group equation, which resembles the Wetterich equation of a bosonic system. As a consequence, the non-trivial structure of the su(2) algebra is fully taken into account by the initial condition of the renormalization group flow. Our method is motivated by the spin-diagrammatic approach to quantum spin system that was developed more than half a century ago in a seminal work by Vaks, Larkin, and Pikin (VLP) [Sov. Phys. JETP 26, 188 (1968)]. By embedding their ideas in the language of the modern renormalization group, we avoid the complicated diagrammatic rules while at the same time allowing for novel approximation schemes. As a demonstration, we explicitly show how VLP's results for the leading corrections to the free energy and to the longitudinal polarization function of a ferromagnetic Heisenberg model can be recovered within the SFRG. Furthermore, we apply our method to the spin-S Ising model as well as to the spin-S quantum Heisenberg model, which allows us to calculate the critical temperature for both a ferromagnetic and an antiferromagnetic exchange interaction. Finally, we present a new hybrid formulation of the SFRG, which combines features of both the pure and the Hubbard-Stratonovich SFRG that were published recently [Phys. Rev. B 99, 060403(R) (2019)].
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.
Great interest has emerged recently in the search for Kitaev spin liquid states in real materials. Such states rely on strongly anisotropic magnetic interactions, which have been suggested to exist in a number of candidate materials based on Ir and Ru. This thesis concentrates on two priority purposes. The first is the investigation of electronic and magnetic properties of candidate materials Na2IrO3, α-Li2IrO3, α-RuCl3, γ-Li2IrO3, and Ba3YIr2O9 for Kitaev physics where both spin-orbit coupling and correlation effects are important. The second is the method development for the microscopic description of correlated materials combining many-body methods and density functional theory (DFT). ...