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arXiv:cond-mat/0107548 · Physical review. B, Condensed matter

Renormalization group for two-dimensional fermions with a flat Fermi surface

Sébastien Dusuel, F. Vistulo de Abreu, Benoît Douçot

Published Feb 5, 2002• 12 citations
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Abstract

We present a renormalization-group analysis of two-dimensional interacting fermion systems with a closed and partially flat Fermi surface. Numerical solutions of the one-loop flow equations show that for a bare local repulsion, the system evolves through three different regimes as the typical energy is lowered: a high-energy transient with a strong competition between particle-particle and particle-hole channels, an intermediate regime with dominant spin-density wave correlations, and finally a ``hot spot'' regime exhibiting d-wave superconductivity. We study, mostly by analytical methods, how this flow pattern depends on the number N of Fermi-surface patches used in the numerical solution. This clearly indicates that the final regime requires vanishingly small microscopic interactions, for the one-loop approximation to be valid, as N is going to infinity.

DOI
10.1103/physrevb.65.094505

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We present a renormalization-group analysis of two-dimensional interacting fermion systems with a closed and partially flat Fermi surface. Numerical solutions of the one-loop flow equations show that for a bare local repulsion, the system evolves through three different regimes as the typical energy is lowered: a high-energy transient with a strong competition between particle-particle and particle-hole channels, an intermediate regime with dominant spin-density wave correlations, and finally a ``hot spot'' regime exhibiting d-wave superconductivity. We study, mostly by analytical methods, how this flow pattern depends on the number N of Fermi-surface patches used in the numerical solution. This clearly indicates that the final regime requires vanishingly small microscopic interactions, for the one-loop approximation to be valid, as N is going to infinity.
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.
We present the formalism for a two-loop renormalization group (RG) calculation of some order-parameter susceptibilities associated with a two-dimensional (2D) flat Fermi-surface model. In this order of perturbation theory, one must take into account the self-energy effects directly in all RG flow equations. In one-loop order, our calculation reproduces the well-known results obtained previously by other RG schemes. That is, for repulsive interactions all susceptibilities diverge in the low-energy limit and the antiferromagnetic (AF) spin-density-wave correlations produce indeed the leading instability in the system. In contrast, in two-loop order, only the AF susceptibility diverges for this model. However, even this divergence takes place at a much slower rate than in the one-loop RG approach. The purpose of this paper is to show in a very simple setting how to assess the importance of two-loop quantum fluctuations in 2D interacting fermionic models. With some modifications, the present formalism can also be extended to discuss more realistic models such as the paradigmatic 2D Hubbard model.
Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofer's renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.
We present a renormalization group (RG) analysis of a fermionic ``hot-spot'' model of interacting electrons on the square lattice. We truncate the Fermi-surface excitations to linearly dispersing quasiparticles in the vicinity of eight hot spots on the Fermi surface, with each hot spot separated from another by the wave vector $(\ensuremath{\pi},\ensuremath{\pi})$. This is motivated by the importance of these Fermi-surface locations to the onset of antiferromagnetic order; however, we allow for all possible quartic interactions between the fermions, and also for all possible ordering instabilities. We compute the RG equations for our model, which depend on whether or not the hot spots are perfectly nested, and relate our results to earlier models. We also compute the RG flow of the relevant order parameters for both Hubbard and J,VJ,VJ,V interactions, and present our results for the dominant instabilities in the nested and non-nested cases. In particular, we find that non-nested hot spots with J,VJ,VJ,V interactions have competing singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting and ddd-form factor incommensurate density wave instabilities. We also investigate the enhancement of incommensurate density waves near experimentally observed wave vectors, and find dominant ddd-form factor enhancement for a range of couplings.
We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the effective low-energy action is generated from the bare action of the microscopic model by a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions, the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities are computed. This and similar RG calculations by others conclusively establish the existence of d-wave superconductivity in the Hubbard model.
We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.