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Modern Physics Letters B

RENORMALIZATION GROUP OF A TWO-DIMENSIONAL PATCHED FERMI SURFACE

A. FERRAZ

Published Feb 20, 2003• 2 citations
Publisher

Abstract

Using the renormalization group we calculate the single particle Green's function G and the momentum occupation function [Formula: see text] for a quasiparticle in a two-dimensional Fermi Surface (FS) composed of four symmetric patches with both flat and curved arcs in [Formula: see text]-space. We show that G develops an anomalous dimension as a result of the vanishing of the quasiparticle weight at the FS. [Formula: see text] is a continuous function of [Formula: see text] with an infinite slope at FS for CU*2/(1 - CU*2) < 1. This result resembles a Luttinger liquid and indicates the breakdown of Fermi liquid theory in this regime.

DOI
10.1142/s021798490300507x

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Using the renormalization group we calculate the single particle Green's function G and the momentum occupation function [Formula: see text] for a quasiparticle in a two-dimensional Fermi Surface (FS) composed of four symmetric patches with both flat and curved arcs in [Formula: see text]-space. We show that G develops an anomalous dimension as a result of the vanishing of the quasiparticle weight at the FS. [Formula: see text] is a continuous function of [Formula: see text] with an infinite slope at FS for CU2/(1 - CU2) < 1. This result resembles a Luttinger liquid and indicates the breakdown of Fermi liquid theory in this regime.
We consider a model composed of Landau quasiparticle states with patched Fermi surfaces (FS) sandwiched by states with flat FS to simulate the "cold" spot regions in cuprates. We calculate the one-particle irreducible function and the self-energy up to two-loop order. Using renormalization group arguments, we show that in the forward scattering channel, the renormalized coupling constant is never infrared stable due to the flat FS sectors. Furthemore, we show that the self-energy scales with energy as Re ∑~ω ln ω as ω→0, and thus the Fermi liquid state within each FS patch is turned into a marginal Fermi liquid.
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 ZZZ along the Fermi surface, and the vanishing of the renormalized coupling functions for several choices of the external momenta.
In the present work, we implement an explicit two-loop renormalization of a two-dimensional flat Fermi surface (FS) in the framework of a field theoretical renormalization group approach (RG). In our scheme, we derive the RG equations for both the coupling functions and Fermi energy. In this way, we are able to probe the existence of spin-charge separation by showing that the low-energy sector of the system is in fact a non-Fermi liquid. In addition, associating the true interacting FS to the infrared stable (IR) fixed point of the Fermi energy, we demonstrate here that it either acquires a small curvature and behaves as a "Luttinger liquid" or it suffers a truncation in k-space depicting an insulating spin liquid.
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.
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.
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.