Using the method of continuous constructive renormalization group around the Fermi surface, we prove that a jellium two-dimensional interacting system of fermions at low temperature T is analytic in the coupling constant lambda for |lambda| |logT|</=K, where K is some numerical constant and T is the temperature. In that range of parameters, the first and second derivatives of the self-energy remain bounded, a behavior seen in Fermi liquids but, in particular, not in Luttinger liquids. Our results also prove that in 2D any transition temperature must be nonperturbative in the coupling constant, a result expected on physical grounds. The proof exploits specific momentum conservation rules in two dimensions.
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.
We give a self-contained derivation of the differential equations for Wilson's renormalization group for the one-particle irreducible Green functions in fermionic systems. The application of this equation to the (t,t′)-Hubbard model appears in Ref. [HSFR]. Here we focus on theoretical aspects. After deriving the equations, we discuss the restrictions imposed by symmetries on the effective action. We discuss scaling properties due to the geometry of the Fermi surface and give precise criteria to determine when they justify the use of one-loop flows. We also discuss the relationship of this approach to other RG treatments, as well as aspects of the practical treatment of truncated equations, such as the projection to the Fermi surface and the calculation of susceptibilities.
We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ4 field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large. We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.
In this paper we complete the first step, namely the uniform bound on completely convergent contributions, towards proving that a three dimensional interacting system of Fermions is a Fermi liquid in the sense of Salmhofer. The analysis relies on a direct space decomposition of the propagator, on a bosonic multiscale cluster expansion and on the Hadamard inequality, rather than on a Fermionic expansion and an angular analysis in momentum space, as was used in the recent proof by two of us of Salmhofer’s criterion in two dimensions.
The naive perturbation expansion for many-fermion systems is infrared divergent. One can remove these divergences by introducing counterterms. To do this without changing the model, one has to solve an inversion equation. We call this procedure Fermi surface renormalization (FSR). Whether or not FSR is possible depends on the regularity properties of the fermion self-energy. When the Fermi surface is nonspherical, this regularity problem is rather nontrivial. Using improved power counting at all orders in perturbation theory, we have shown sufficient differentiability to solve the FSR equation for a class of models with a non-nested, non-spherical Fermi surface. I will first motivate the problem and give a definition of FSR, and then describe the combination of geometric and graphical facts that lead to the improved power counting bounds. These bounds also apply to the four-point function. They imply that only ladder diagrams can give singular contributions to the four-point function.