We consider a system of two-dimensional interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non-Luttinger liquid behaviors found through different approximations is resolved by showing the existence of a line of nontrivial fixed points, for the renormalization group (RG) flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is nonperturbative and based on the implementation, at each RG iteration, of Ward identities obtained from local phase transformations depending on the Fermi surface side, implying the partial vanishing of the beta function.
We implement an explicit two-loop calculation of the coupling functions and the self-energy of interacting fermions with a two-dimensional flat Fermi surface in the framework of the field theoretical renormalization group (RG) approach. Throughout the calculation both the Fermi surface and the Fermi velocity are assumed to be fixed and unaffected by interactions. We show that in two dimensions, in a weak coupling regime, there is no significant change in the RG flow compared to the well-known one-loop results available in the literature. However, if we extrapolate the flow to a moderate coupling regime there are interesting new features associated with an anisotropic suppression of the quasiparticle weight
Z along the Fermi surface, and the vanishing of the renormalized coupling functions for several choices of the external momenta.
Using perturbation theory and the field theoretical renormalization group approach we consider a two-dimensional anisotropic truncated Fermi Surface (FS) with both flat and curved sectors which approximately simulates the ``cold'' and ``hot'' spots in the cuprate superconductors. We calculate the one-particle two-loop irreducible functions ${\ensuremath{\Gamma}}^{(2)}$ and ${\ensuremath{\Gamma}}^{(4)}$ as well as the spin, the charge and pairing response functions up to one-loop order. We find nontrivial infrared stable fixed points and we show that there are important effects produced by the mixing of the existing scattering channels in higher order of perturbation theory. Our results indicate that the ``cold'' spots are turned into a non-Fermi liquid with divergents $\ensuremath{\partial}{\ensuremath{\Sigma}}_{0}/\ensuremath{\partial}{p}_{0}$ and $\ensuremath{\partial}{\ensuremath{\Sigma}}_{0}/\ensuremath{\partial}\overline{p},$ a vanishing Z, and either a finite or zero ``Fermi velocity'' at the FS when the effects produced by the flat portions are taken into account.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.
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 analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension $\ensuremath{\eta}(\mathbf{p})$ which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi-liquid (NFL) regime at temperature $T\ensuremath{\rightarrow}0$ displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing susceptibility finally produces the dominant instability in the low-energy limit.