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arXiv:cond-mat/9609118 · Physical review. B, Condensed matter

Parquet solution for a flat Fermi surface

Anatoley T. Zheleznyak, Victor M. Yakovenko, I. E. Dzyaloshinskiǐ

Published Feb 1, 1997• 123 citations
arXivHTMLPDF

Abstract

We study instabilities occurring in the electron system whose Fermi surface has flat regions on its opposite sides. Such a Fermi surface resembles Fermi surfaces of some high-${\mathrm{T}}_{\mathrm{c}}$ superconductors. In the framework of the parquet approximation, we classify possible instabilities and derive renormalization-group equations that determine the evolution of corresponding susceptibilities with decreasing temperature. Numerical solutions of the parquet equations are found to be in qualitative agreement with a ladder approximation. For the repulsive Hubbard interaction, the antiferromagnetic (spin-density-wave) instability dominates, but when the Fermi surface is not perfectly flat, the d-wave superconducting instability takes over.

DOI
10.1103/physrevb.55.3200

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We study instabilities occurring in the electron system whose Fermi surface has flat regions on its opposite sides. Such a Fermi surface resembles Fermi surfaces of some high-Tc{\mathrm{T}}_{\mathrm{c}}Tc​ superconductors. In the framework of the parquet approximation, we classify possible instabilities and derive renormalization-group equations that determine the evolution of corresponding susceptibilities with decreasing temperature. Numerical solutions of the parquet equations are found to be in qualitative agreement with a ladder approximation. For the repulsive Hubbard interaction, the antiferromagnetic (spin-density-wave) instability dominates, but when the Fermi surface is not perfectly flat, the d-wave superconducting instability takes over.
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 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.
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We investigate a mechanism of superconductivity on the basis of the nested-Fermi-liquid theory. A set of parquet equations is derived to investigate the instability towards superconductivity and spin- or charge-density-wave states. The phase diagram is examined on the basis of the parquet equations. We find that the polarization mode of the marginal-Fermi-liquid-type is of vital importance to superconductivity. We also discuss the properties of a superconductor on the basis of the nested Fermi liquid.
We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.
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.