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arXiv:cond-mat/0203376 · Europhysics Letters (EPL)

Fermi surface renormalization in two spatial dimensions

Álvaro Antônio Bandeira Ferraz

Published Jan 1, 2003• 8 citations
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Abstract

We discuss the renormalization induced by interactions of a two-dimensional truncated Fermi surface (FS) model. Using a field-theoretical renormalization group method, we calculate the critical renormalized physical chemical potential μ R (μ/ω; U (ω)) which may take different values along the Fermi surface. We show that FS is strongly renormalized by interactions in two dimensions. As a result of that, FS can be further truncated in those k -space regions where non-trivial fixed points produce a smooth momentum distribution function in the vicinity of the Fermi surface.

DOI
10.1209/epl/i2003-00223-5

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We discuss the renormalization induced by interactions of a two-dimensional truncated Fermi surface (FS) model. Using a field-theoretical renormalization group method, we calculate the critical renormalized physical chemical potential μ R (μ/ω; U (ω)) which may take different values along the Fermi surface. We show that FS is strongly renormalized by interactions in two dimensions. As a result of that, FS can be further truncated in those k -space regions where non-trivial fixed points produce a smooth momentum distribution function in the vicinity of the Fermi surface.
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 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.
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.
We study the non-Fermi-liquid state formed by an isotropic, degenerate Fermi gas in two spatial dimensions interacting with a U(1) gauge field. Our calculation uses the functional renormalization group (fRG) with a soft frequency cutoff for the fermions. The fRG scheme we employ takes account of the gauge symmetry, which imposes relations (modified Ward-Takahashi identities) between the couplings which constrain the RG flow. The critical exponents and couplings we find for the resulting non-Fermi liquid are mostly insensitive to whether we enforce the gauge symmetry constraints, which signifies either that the constraints are superfluous or that the frequency-cutoff scheme is particularly robust. The exception is the gauge-boson mass term, which is RG-relevant about the fixed point without the constraints but is irrelevant when they are enforced. The latter is physically accurate, as a gauge symmetry cannot be spontaneously broken. In addition, we find z = 2 and Σ ( ω , k F ) ∼ ω 1 / 2 for the boson dynamical exponent and scaling of the fermion self-energy, respectively. These results differ considerably from those of past works on the model, though we argue for their plausibility.
We devise a dimensional regularization scheme for quantum field theories with a Fermi surface to study scaling behavior of non-Fermi liquid states in a controlled approximation. Starting from two space dimensions, the co-dimension of the Fermi surface is extended to a general value while its dimension is fixed. When the Fermi surface is coupled with a critical boson centered at zero momentum, the interaction becomes marginal at a critical space dimension dc=5/2{d}_{c}=5/2dc​=5/2. A deviation from the critical dimension is used as a small parameter for a systematic expansion. We apply this method to the theory where two patches of the Fermi surface are coupled with a critical boson, and show that the Ising-nematic critical point is described by a stable non-Fermi liquid state slightly below the critical dimension. Critical exponents are computed up to the two-loop order.
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.