We present a detailed study of thermal transport in the disordered Fermi liquid with short-range interactions. At temperatures smaller than the impurity scattering rate, i.e., in the diffusive regime, thermal conductivity acquires nonanalytic quantum corrections. When these quantum corrections become large at low temperatures, the calculation of thermal conductivity demands a theoretical approach that treats disorder and interactions on an equal footing. In this paper, we develop such an approach by merging Luttinger's idea of using gravitational potentials for the analysis of thermal phenomena with a renormalization group calculation based on the Keldysh nonlinear sigma model. The gravitational potentials are introduced in the action as auxiliary sources that couple to the heat density. These sources are a convenient tool for generating expressions for the heat density and its correlation function from the partition function. Already in the absence of the gravitational potentials, the nonlinear sigma model contains several temperature-dependent renormalization group charges. When the gravitational potentials are introduced into the model, they acquire an independent renormalization group flow. We show that this flow preserves the phenomenological form of the correlation function, reflecting its relation to the specific heat and the constraints imposed by energy conservation. The main result of our analysis is that the Wiedemann-Franz law holds down to the lowest temperatures even in the presence of disorder and interactions and despite the quantum corrections that arise for both the electric and thermal conductivities.
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.
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.
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.
We construct exact functional renormalization group (RG) flow equations for nonrelativistic fermions in arbitrary dimensions, taking into account not only mode elimination, but also the rescaling of the momenta, frequencies, and fermionic fields. The complete RG flow of all relevant, marginal, and irrelevant couplings can be described by a system of coupled flow equations for the irreducible n-point vertices. Introducing suitable dimensionless variables, we obtain flow equations for generalized scaling functions which are continuous functions of the flow parameter, even if we consider quantities which are dominated by momenta close to the Fermi surface, such as the density-density correlation function at long wavelengths. We also show how the problem of constructing the renormalized Fermi surface can be reduced to the problem of finding the RG fixed point of the irreducible two-point vertex at vanishing momentum and frequency. We argue that only if the degrees of freedom are properly rescaled is it possible to reach scale-invariant non-Fermi-liquid fixed points within a truncation of the exact RG flow equations.