The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.
The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.
We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.
We investigate nonequilibrium properties of the single-impurity Anderson model by means of the functional renormalization group (fRG) within Keldysh formalism. We present how the level broadening $\ensuremath{\Gamma}/2$ can be used as flow parameter for the fRG. This choice preserves important aspects of the Fermi-liquid behavior that the model exhibits in case of particle-hole symmetry. An approximation scheme for the Keldysh fRG is developed which accounts for the frequency dependence of the two-particle vertex in a way similar but not equivalent to a recently published approximation to the equilibrium Matsubara fRG. Our method turns out to be a flexible tool for the study of weak to intermediate on-site interactions $U\ensuremath{\lesssim}3\ensuremath{\Gamma}$. In equilibrium we find excellent agreement with numerical RG results for the linear conductance at finite gate voltage, magnetic field, and temperature. In nonequilibrium, our results for the current agree well with time-dependent density-matrix RG. For the nonlinear conductance as function of the bias voltage, we propose reliable results at finite magnetic field and finite temperature. Furthermore, we demonstrate the exponentially small scale of the Kondo temperature to appear in the second-order derivative of the self-energy. We show that the approximation is, however, not able to reproduce the scaling of the effective mass at large interactions.
Technological advances in the fabrication of mesoscopic structures allow for an experimental investigation of low-dimensional interacting many-body systems in nonequilibrium situations. From a theoretical perspective, the accurate description of such systems constitutes a major challenge which has in general not been solved. In light of possible applications, such as quantum computation, but also for a better understanding of correlation effects themselves, it is thus important to put a research focus on methodological advances.The so called functional renormalization group (fRG) has turned out to be a successful and versatile tool in the application to the aforementioned systems. Being an RG method, it has the associated advantages, e.g. a successive integrating out of energy scales and avoiding divergences that perturbation theory would have. In this thesis, we present two paradigmatic extensions of the fRG approach.Typically, the fRG is used to investigate the flow of the generating functional of the one-particle-irreducible (1PI) vertex functions. Typical approximations in this framework bring about certain shortcomings, such as the violation of conservation laws. In order to overcome some of these shortcomings, we formulate the fRG for the generating functional of the two-particle-irreducible (2PI) vertex functions. This step constitutes our first extension of the fRG approach. While typically only nonconserving schemes are generated in 1PI fRG, we can define nonconserving and conserving approximation schemes within 2PI fRG. Partially, these are found to be equivalent to known diagrammatic approximations. We apply the various 2PI fRG schemes to the quantum anharmonic oscillator, as a “toy model”, and to the Anderson impurity model, as a full fermionic quantum many-body problem. Overall, we find that the devised 2PI fRG schemes (even the conserving ones) offer no great advantage over established 1PI methods. As an exception, the nonconserving lowest-order 2PI fRG scheme in which the flow parameter is introduced in the interaction produces better results than 1PI fRG schemes which are comparable in computational effort.Our second extension of the fRG approach is to apply it to models which cannot (directly) be formulated in second quantization. Having a localized spin, the Kondo model is a simple example of such models which nevertheless shows nontrivial (but well-understood, e.g. by Anderson’s Poor Man’s Scaling) correlation effects. It has hardly been treated with fRG in the literature. In this thesis, we present how to overcome obstacles arising in the formulation of the Kondo model in functional integral formalism. Instead of the 1PI fRG, we use the Polchinski fRG which is based on connected amputated Green functions. This makes it possible to make a close connection to Poor Man’s Scaling which employs the concept of the T-matrix. We reproduce the results of Poor Man’s Scaling and also devise a systematic extension to it by keeping the frequency dependence of the flowing amputated Green functions in the RG equations. With this scheme, we do not find a significant qualitative improvement over Poor Man’s Scaling. However, we have layed the groundwork for more successful future extensions of the fRG applied to the Kondo model.
This thesis comprises two parts centered around the functional renormalization-group framework: in the first part, I study the role of symmetries and conservation laws in approximate solutions, while in the second part I analyze Friedel oscillations and transport in Luttinger liquids with impurities. The functional renormalization group (fRG) has been developed as a new computational tool in the theory of interacting Fermi systems. The effective behavior of a given microscopic model is calculated by solving coupled differential flow equations for the Green functions with an energy scale as the flow parameter. The symmetries of the microscopic model imply Ward identities between Green and response functions. It is shown that solutions of truncated flow-equation hierarchies satisfy Ward identities if the cutoff bare action is gauge invariant. However, truncations are generally not self-consistent approximations in the sense of Baym and Kadanoff. The fRG is then applied to study Luttinger liquids. By computing the full spatial effective potential of a single impurity, long-range Friedel oscillations are observed in the density profile with the expected power laws for systems with up to 10^7 lattice sites. For a double barrier enclosing a dot region we find temperature regimes in which the conductance follows power laws with universal exponents, as well as non-universal crossover regimes in intermediate parameter regions.
The experimental progress in cold atomic gases and mesoscopic systems has allowed unprecedented access to low dimensional correlated quantum systems far from equilibrium. This has revealed a variety of novel phenomena such as many-body localization, time crystals and dynamical phase transitions. The theoretical description of such systems has remained a challenge: The numerically exact solution of generic, correlated quantum systems requires computational effort that scales exponentially in the number of fermions, rendering larger systems and therefore collective behavior inaccessible. This necessitates approximations. In low dimensions, infrared divergences prohibit the use of a purely perturbative approach. Renormalization group based techniques, where energy scales are treated successively remedy some of these shortcomings. We therefore propose the use of the functional renormalization group to access correlation functions far from equilibrium in large, interacting systems. Such techniques were previously successfully applied in the investigation of quantum impurity problems and quenches in tight-binding chains and are widely used in the study of equilibrium phase diagrams. After a presentation of the Keldysh formalism, that constitutes a natural language out of equilibrium, we present perturbation theory and the functional renormalization group approach in a diagrammatic language. We then develop, optimize and apply flexible fRG techniques to four different problems in one-dimensional tight-binding chains with short-ranged two-particle interaction. First, we discuss the emergence of anomalous transport in the presence of a quasiperiodic potential by analyzing linear response and a far-from-equilibrium quench protocol. We then develop a variant of the fRG to access pure excited eigenstates. We use this algorithm to investigate many-body states in large tight-binding chains and show that generic excitations appear thermal while some states show unique spectral properties. To study the influence of inelastic scattering on the transport properties in such a chain we then go beyond an effective single-particle picture; to that end, we present an algorithm, that includes scattering in long chains. Finally, we discuss infinite chains coupled to a substrate while driven out of equilibrium by an electric field. Exploiting the inherent symmetries, we develop efficient algorithms and apply them to a simple tight-binding chain with nearest-neighbor interaction to explore the interplay of spontaneous symmetry breaking and driving. We demonstrate, that ordering is suppressed and provide results that indicate unconventional transport in the driven system.
The aim of this work is the extension of the functional renormalization group (FRG) formalism to treat non-equilibrium situations. To this end, we reformulate the FRG equations in terms of the Keldysh method which is the standard technique to treat systems out of equilibrium. As simplest non-trivial application to test the potential and weakness of the non-equilibrium FRG we choose the single impurity Anderson model (SIAM). This model represents the paradigm for correlation effects in condensed matter physics and it is at the heart of a large range of experimental and theoretical investigations. In particular, the SIAM can be considered as the standard model for describing the physical properties of certain nanostructures and mesoscopic systems, such as quantum dots. These devices are interesting because they show fancy physical effects such as single electron tunneling and they could open new perspectives for future generations of electronic devices. After a general introduction to mesoscopic systems and the basics of the Keldysh technique, we derive the FRG equations for treating non-equilibrium situations and we point out the differences between equilibrium and non-equilibrium FRG schemes. The FRG can be adopted to describe both fermionic and bosonic systems and, at least formally, time-dependent situations as well as the stationary case. In the second part of this work we analyze the transport properties of quantum dots in several physical situations. First, we study the ""easiest"" case, namely we apply the non-equilibrium FRG formalism to the SIAM with finite bias voltage V_B in the stationary state at T=0. Here, we can test the technique by comparing it to available exact results for the linear response regime V_B = 0. As next step we switch on an external magnetic field B and the temperature T in order to observe the effect of these two parameters on the transport properties of a quantum dot. The comparison to known results for V_B = 0 allows us to critically survey the range of applicability of the Non-equilibrium FRG and the accuracy of the results.