The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A
d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function
F, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.
After its first isolation by Geim and Novoselov in 2004, graphene has grown quickly into an independent subfield of contemporary condensed matter physics, counting thousands of publications. This material has attracted so much attention not only because it is truly two-dimensional -- a feature long thought to be impossible to be realized in nature -- but also because it is the prime representative of a new class of condensed matter systems, the Dirac materials, whose electronic spectrum disperses linearly around isolated points in the Brillouin zone. An outstanding consequence of this linear energy-momentum dispersion is the collapse of the Fermi surface to isolated points at charge neutrality, accompanied by a vanishing single-particle density of states. While this feature of the noninteracting model is what separates graphene from most other, more conventional condensed matter systems, giving rise to many of its astounding physical properties, it leads to severe computational complications when extending the model to contain two-body interactions and/or disorder. In this thesis we consider both extensions separately. After rigorously deriving the low-energy quantum field theory from a tight-binding model and establishing the one-particle irreducible (1PI) vertex functions as the fundamental building blocks of correlation functions, we construct a nonperturbative formalism for the calculation of these vertex functions using the well-established functional renormalization group (fRG). By combining the fRG with the Keldysh formalism, accounting for possible external electromagnetic fields and nontrivial initial correlations, this framework is capable of handling thermal equilibrium and true nonequilibrium alike. To circumvent some of the technical problems of the standard fRG formalism when treating fermionic systems at finite density, we also explore a variant of the fRG, where the chemical potential is interpreted as a flow parameter. We obtain hierarchical sets of flow equations, which describe the change of the 1PI vertex functions upon varying an artificial cutoff scale or the chemical potential, respectively. These nonperturbative formalisms have been used to calculate the renormalization of the Fermi velocity and the static dielectric function at finite temperature and density in the strong coupling regime as an explicit demonstration of their capabilities. Since the Keldysh fRG is an exact formulation of quantum field theory, in particular when external magnetic fields are present, it should be possible to describe the fractional quantum Hall effect in this framework. However, the typical truncation schemes that are currently available to approximately solve the fRG flow equation fail to account for the nontrivial correlation effects necessary to access the fractional quantum Hall regime. To circumvent this problem we consider a modified field theory, where these correlations are implemented ``by hand'' via a Chern-Simons gauge field, coupling to the fermions in addition to the Coulomb interaction term. This modified field theory is analyzed in a stationary phase approximation including Gaussian fluctuations. The electromagnetic response tensor is calculated in the random phase approximation, yielding the Hall conductivities. In the second part of this thesis we consider disordered Dirac fermions in the absence of two-particle interactions. After sketching how the most general disorder potential for Dirac fermions arises from a tight-binding model, we first consider an explicit physical scenario, a disordered graphene pn junction in the presence of a quantizing magnetic field perpendicular to the graphene sheet. We derive an effective one-dimensional theory for the chiral states that propagate along the junction interface and calculate the full conductance distribution in the crossover between the clean and the strong-disorder limit, via an exact solution of a Fokker-Planck equation. Lastly, we develop a nonperturbative approach to calculate the disorder induced self-energy that is based on exact Schwinger-Dyson equations and Ward identities, not only for graphene, but also for other paradigmatic semimetals with a nodal point dispersion.
We solve the problem of weakly disordered interacting fermions near the flat two-dimensional Fermi surface. For this problem we develop a new angle resolved renormalization group theory that takes into account both the interactions and the disorder as functions of angular position of the electrons near the Fermi surface. The model is treated by the N-patch renormalization group procedure upon the effective action for the partition function for M replicas. The flow equation for the effective interaction and for the effective diffusion function (of three angles) is integrated numerically and the phase diagram is constructed. The low temperature phase is associated to the dominant eigenvalue of the diverging susceptibility. The angular dependence of the corresponding eigenvectors determines the form and the symmetry of the order parameter. The simultaneously diverging diffusion function D provides information about the effects of disorder to the low-temperature state.
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 consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}
vF kF /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.
We consider systems of nonrelativistic, interacting electrons at finite density and zero temperature in d=2,3,ldots dimensions. Our main concern is to characterize those systems that, under the renormalization flow, are driven away from the Landau Fermi-liquid (LFL) renormalization-group fixed point. We are especially interested in understanding under what circumstances such a system is a marginal Fermi-liquid (MFL) when the dimension of space is d\ensuremath{\geqslant}2. The interacting electron system is analyzed by combining renormalization-group (RG) methods with so called 'Luther-Haldane' bosonization techniques. The RG calculations are organized as a double expansion in the inverse scale parameter ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ , which is proportional to the width of the effective momentum space around the Fermi surface and in the running coupling constant ${\mathrm{g}}_{\ensuremath{\lambda}}$ , which measures the strength of electron interactions at energy scales \ensuremath{\sim}
vF kF /\ensuremath{\lambda}. For systems with a strictly convex Fermi surface, superconductivity is the only symmetry-breaking instability. Excluding such an instability, the system can be analyzed by means of bosonization. The RG and the underlying perturbation expansion in powers of ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$ serve to characterize the approximations involved by bosonizing the system. We argue that systems with short-range interactions flow to the LFL fixed point. Within the approximations involved by bosonization, the same holds for systems with long-range, longitudinal, density-density interactions. For electron systems interacting via long-range, transverse, current-current interactions, a deviation from LFL behavior is possible: if the exponent \ensuremath{\alpha} parametrizing the singularity of the interaction potential in momentum space by V-hat(|p|)\ensuremath{\sim}1/|p${\mathrm{|}}^{\mathrm{\ensuremath{\alpha}}}$ is greater than or equal to d-1, the results of the bosonization calculation are consistent with a MFL.