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Annalen der Physik

Weakly interacting electrons and the renormalization group

Benedikt Binz, D. Baeriswyl, Benoît Douçot

Published Nov 14, 2003• 0 citations
Publisher

Abstract

Abstract We present a general method to study weak‐coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high‐energy spin fluctuations create an effective attraction between the quasi‐particles. As an example, we show the occurrence of d ‐wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one‐loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one‐loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two‐dimensional Fermi system at Van Hove filling without nesting.

DOI
10.1002/andp.200351511-1204

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Abstract We present a general method to study weak‐coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high‐energy spin fluctuations create an effective attraction between the quasi‐particles. As an example, we show the occurrence of d ‐wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one‐loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one‐loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two‐dimensional Fermi system at Van Hove filling without nesting.
We present a general method to study weak-coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. As an example, we show the occurrence of d-wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one-loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one-loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two-dimensional Fermi system at Van Hove filling without nesting.
We present a systematic weak-coupling renormalization group (RG) technique for studying a collection of NNN coupled one-dimensional interacting electron systems, focusing on the example of NNN-leg Hubbard ladders. For N=2,3,N=2,3,N=2,3, we recover previous results, and find that also more generally broad regions of the phase space of these models are unstable to pairing, usually with approximate ddd-wave symmetry. We show how these instabilities can be understood in terms of a fairly conventional ``gap'' function \ensuremath{\Delta} at the discretized Fermi surface, and describe how this function is calculated. The dimensional crossovers as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and as many such ladders are weakly coupled together are also discussed.
The problem of weakly correlated electrons on a square lattice is formulated in terms of a one-loop renormalization group. Starting from the action for the entire Brillouin zone we reduce successively the cutoff \ensuremath{\Lambda} about the Fermi surface and follow the renormalization of the coupling U as a function of three energy-momenta. We calculate the intrinsic temperature scale Tco{\mathit{T}}_{\mathit{co}}Tco​ where the renormalization-group flow crosses over from the regime (\ensuremath{\Lambda}>Tco{\mathit{T}}_{\mathit{co}}Tco​) where the electron-electron (e-e) and electron-hole (e-h) terms are equally important to the regime (\ensuremath{\Lambda}Tco{\mathit{T}}_{\mathit{co}}Tco​) where only the e-e term plays a role. In the low-energy regime only the pairing interaction V is marginally relevant, containing contributions from all renormalization-group steps of the regime \ensuremath{\Lambda}>Tco{\mathit{T}}_{\mathit{co}}Tco​. We identify the most attractive eigenvalue ${\ensuremath{\lambda}}_{\mathrm{min}}$ of ${\mathit{V}}_{\mathrm{\ensuremath{\Lambda}}={\mathit{T}}_{\mathit{co}}}$. At low filling, ${\ensuremath{\lambda}}_{\mathrm{min}}$ corresponds to the B2{\mathit{B}}_{2}B2​ representation (dxy{\mathit{d}}_{\mathit{xy}}dxy​ symmetry), while near half filling the strongest attraction occurs in the B1{\mathit{B}}_{1}B1​ representation (${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$ symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. We also give a possible interpretation of angle-resolved photoemission spectroscopy experiments trying to determine the symmetry of the order parameter in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compound Bi2{\mathrm{Bi}}_{2}Bi2​Sr2{\mathrm{Sr}}_{2}Sr2​CaCu2{\mathrm{CaCu}}_{2}CaCu2​O8{\mathrm{O}}_{8}O8​. \textcopyright{} 1996 The American Physical Society.
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 describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from 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 signalled by the diverging interactions we compute the flow of 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 antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.
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.